Thanks Peto for an enjoyable crossword. Definitions are underlined in the clues.
Across
1 Directed towards back street to get unusual kind of medicine (7)
NOSTRUM : Reversal of(… back) ON(directed towards/aimed at …, as in “attacks on his integrity”) + ST(abbrev. for “street”) plus(to get) RUM(unusual/odd).
Defn: … prepared by an unqualified person.
5 Game, unfinished one, about to keep John playing (7)
MAHJONG : Reversal of(…, about) “game”(one, referring to the definition in the clue) minus
its last letter(unfinished …) containing(to keep) anagram of(… playing) JOHN.
9 Small instrument for cutting (5)
SHARP : S(abbrev. for “small”) + HARP(a musical instrument).
10 Photo capturing Heather with South American catapult (9)
SLINGSHOT : SHOT(a still photo) containing(capturing) [LING(heather, the dominant flowering plant in most European heathland) + S(abbrev. for “south”)].
Defn: North American term for a catapult.

11 Extremely good for former partner to find a girl in Connecticut (9)
EXCELLENT : EX(one’s former partner in an intimate relationship) plus(to find) [ELLEN(a girl’s name) contained in(in) CT(abbrev. for Connecticut, the US state)].
12 Instruction for players from City – “Try accepting a bit of responsibility” (5)
LARGO : LA(abbrev. for Los Angeles, the US city) + GO(a try/an attempt) containing(accepting) 1st letter of(a bit of) “responsibility“.
Defn: …/musicians to play slowly and in a dignified style.
13 Heard gossip on the way back implicating earl (5)
TRIED : Reversal of(… on the way back) DIRT(gossip about someone that could be damaging if revealed) containing(implicating) E(abbrev. for “earl”).
Defn: … in a court of law.
15 Taunt in pub gets social worker justifiably angry (9)
INDIGNANT : DIG(a taunt/a mocking remark) contained in(in) INN(a pub) plus(gets) ANT(a social insect, perhaps one that belongs to the worker caste).
18 Tory uncertainty brought about negative effect for sure (9)
CONFIDENT : CON(abbrev. for a member of the Conservative Party, a Tory) + reversal of(… brought about) IF(an uncertainty/a supposition, as in “there are too many ifs in the proposal”) + DENT(a negative effect/a little damage, as in “he suffered a dent to his self-confidence”).
19 Wife taken in by a trace of local intonation (5)
TWANG : W(abbrev. for “wife”) contained in(taken in by) TANG(a trace/a suggestion of).
21 After an awkward situation May oddly gets lucky (5)
JAMMY : JAM(an awkward/difficult situation to be in) + 1st and 3rd letters of(… oddly) “May“.
Defn: … in British slang. Did she get lucky?
23 Almost certain to aim at the queen after king’s capitulation (9)
SURRENDER : “sure”(certain/definite) minus its last letter(Almost …) plus(to) [END(an aim/an objective) plus(at) [ER(abbrev. for Elizabeth Regina, the queen) placed after(after) R(abbrev. for “Rex”, king in Latin)].
25 Vital cure not exactly profitable (9)
LUCRATIVE : Anagram of(… not exactly) VITAL CURE.
26 Country in Africa principally associated with secret society (5)
TONGA : 1st letter of(… principally) “Africa” placed after(associated with) TONG(a Chinese secret society often associated with organised crime).
Defn: … not in Africa but in Polynesia.
27 Extremely stoical about lawyer facing jail for slander (7)
SCANDAL : 1st and last letters of(Extremely) “stoical” containing(about) [DA(abbrev. for “District Attorney”, a US lawyer) placed after(facing) CAN(slang for jail/prison)].
28 Tad lewd novel’s pretentious writing (7)
TWADDLE : Anagram of(… novel) TAD LEWD.
Down
1 Starting to develop a hint of nastiness prior to ascent (7)
NASCENT : 1st letter of(a hint of) “nastiness” plus(prior to) ASCENT.
2 Sanction changes to cover husband’s means of support (9)
STANCHION : Anagram of(… changes) SANCTION containing(to cover) H(abbrev. for “husband”).

3 Drive back from dance without Penny (5)
REPEL : REEL(a lively Scottish or Irish folk dance) containing(without) P(abbrev. for “penny”, in British currency).
4 Transfix my French setter with an increase in sound (9)
MESMERIZE : MES(French for “my” as in “mes amis et moi”) + ME(self-referential pronoun for the setter of this puzzle) plus(with) homophone of(… in sound) “rise”(an increase, in quantity, say).
The clue also works with the non-American spelling of the solution and without the homophone.
5 Damp from fine spray carrying over (5)
MOIST : MIST(a fine spray of tiny droplets) containing(carrying)O(abbrev. for “over” in cricket scores).
6 Point out frivolous noble earlier (9)
HIGHLIGHT : LIGHT(frivolous/without much serious purpose or value) placed below(… earlier, in a down clue) HIGH(noble/of superior rank or status).
7 Additional concern after losing opener (5)
OTHER : “bother”(to feel concern for/to take interest in, as in “don’t bother about him; he can do whatever he wants”) minus its 1st letter(after losing opener).
Defn: …/more, as in “my other concern is …”
8 Clear off with contract unable to be found (3,4)
GET LOST! : GET(to contract, as with a contagious disease) + LOST(unable to be found/gone astray).
14 Why speaking with Maria did wrong by old farm worker (9)
DAIRYMAID : Anagram of(… wrong) [Homophone of(… speaking) “why” plus(with) MARIA DID].
The old and the not so old:

16 Put off by spot of grease on hospital department cleaner (9)
DETERGENT : DETER(to put off/to discourage) plus(by) 1st letter of(spot of) “grease” placed above(on, in a down clue) ENT(abbrev. for the Ear, Nose and Throat specialist department in a hospital)
17 Uninhibited publicity material featuring single by pop group (9)
ABANDONED : AD(short for “advertisement”, promotional/publicity material) containing(featuring) [ONE(single) placed below(by, in a down clue) BAND(a pop group)].
18 Coaxes Jack and Bill to return with shouts of encouragement (7)
CAJOLES : Reversal of(… to return) [J(abbrev. for “Jack”, one of 4 in a deck of playing cards) plus(and) AC(abbrev. for “account”, a statement of what’s due and what’s paid/a bill)] plus(with) OLÉS(shouts of encouragement, for an exceptional performance, often associated with bullfighting).
20 Relevant to treatment of men with rage (7)
GERMANE : Anagram of(treatment of) [MEN plus(with) RAGE].
22 Mao essentially ignored tea and coffee (5)
MOCHA : “Mao” minus its middle letter(essentially ignored) + CHA(tea).
23 Not sparkling yet (5)
STILL : Double defn: 1st: …, as with bottled mineral water; and 2nd: …, as in “the best is yet to come”.
24 Run without the right attitude at first (5)
EXTRA : EX-(prefix signifying “without”/not with, as in “a share price ex-dividend”) + 1st letters, respectively, of(… at first) “the right attitude“.
Defn: In cricket, a run scored other than from a hit with the bat, credited to the team, rather than to the batsman.
For 23A, you have the chess abbreviation for King. It should have been the Latin abbreviation
Thanks aid.
Thanks to Peto and scchua. Very enjoyable. I did not know the slang JAMMY but the parsing was clear and I was slow in parsing CONFIDENT thanks to the reversed “if.”
There are times that I have more trouble than I should have, solving Peto’s (or Poins’s) crosswords.
I think that is mainly because he uses a lot of what we call juxtaposition indicators.
However, today’s went it smoothly.
All well clued but I was a bit surprised by 1d: ‘ascent’ in the clue, and part of the solution (NASCENT).
Must have been an oversight by Peto (who, I think, wouldn’t usually do this).
Many thanks to the setter (and looking forward to see you again in York) & scchua (I still remember meeting you at the first Derby S&B, ages ago)
@Sil van den Hoek— I too was surprised by the giveaway cluing in 1d. I would have at least expected something like “starting to develop sodium odour.” Nonetheless, thanks Peto for an enjoyable crossword and Scchua for parsing.
Thanks Peto and scchua
Entertaining puzzle that didn’t have to many hold ups in it. Funny about 1d, as a ‘giveaway’ clue, it did not get filled in until the latter part of the solve – a classic case of hiding the word play blatantly !
JAMMY was a new term for me and the spelling of STANCHION had me thinking for a little while – couldn’t get the U out of my head.
MOCHA was the first in CONFIDENT and TRIED the last couple in.