A good Quiptic, with the NE corner the last to yield. Thanks to Carpathian. Definitions are underlined in the clues.
Across
1 Swimmer private investigator managed to hoax regularly (7)
PIRANHA : PI(abbrev. for “private investigator”/private eye) + RAN(managed, as with an organisation) plus(to) 1st and 2nd letters of(… regularly) “hoax“.

5 Believes in maintaining hard drives (7)
THRUSTS : TRUSTS(believes in/accepts as reliable) containing(maintaining) H(abbrev. for “hard”).
9 Religious instruction to the German jockey (5)
RIDER : RI(abbrev. for “religious instruction”) plus(to) DER(German for “the”).
10 Monument of leader’s character (9)
HEADSTONE : HEAD(leader/headman)‘S + TONE(general character/quality of a place, piece of writing, situation, etc.)
11 Regard card head shows trial participant (10)
EYEWITNESS : EYE(to regard with, well, one’s eyes) + WIT(a person who is amusing/a card) + NESS(a head/a headland/a promontory).
12 Just fine atmosphere (4)
FAIR : F(abbrev. for “fine”) + AIR(atmosphere/the space above earth’s surface).
14 Ruins ground thrown all about (11)
SURROUNDING : Anagram of(… thrown) RUINS GROUND.
18 Number of article relating to a sense of beauty (11)
ANAESTHETIC : AN(an article in grammar) + AESTHETIC(relating to a sense of beauty/appreciation for beauty).
Defn: Numb-er/a numbing agent.
21 Exposed and inwardly vulnerable when retreating (4)
BARE : Hidden in(inwardly) reversal of(… when retreating) “vulnerable“.
22 Redesign planetarium without right engineer (10)
MANIPULATE : Anagram of(Redesign) “planetarium” minus(without) “r”(abbrev. for “right”).
25 Air assistance coming back with one second opinion (9)
DIAGNOSIS : Reversal of(… coming back) [ SONG(air/a tune) + AID(assistance/help) ] plus(with) I(Roman numeral for “one”) + S(abbrev. for “second” in time notation).
26 One knocking top off large fireplace (5)
INGLE : “single”(only one) minus its 1st letter(knocking top off).
Defn: … in the home – does it have to be large?
27 Tracked part of play over time with director (7)
SCENTED : SCENE(part of a play on stage) containing(over) T(abbrev. for “time”) plus(with) D(alternative to “Dir.”/abbrev. for “director”). D0 As a verb, recognised/detected by the sense of smell.
28 Approve goal men seek regularly (7)
ENDORSE : END(a goal/an objective, as in “a means to an end”) + OR(abbrev. for “other ranks”/military men other than commissioned officers) + 1st and 3rd letters of(… regularly) “seek“.
Down
1 Penny pressed and cleaned (6)
PURGED : P(abbrev. for “penny”, a unit of currency) + URGED(pressed/tried to persuade someone to do something).
Defn: Got rid of something unwanted or removed undesirable people from a place or organisation.
2 Left European space in exchange for money (6)
REDEEM : RED(colour associated with left-wing politics) + E(abbrev. for “European”) + EM(a unit of space between characters in typesetting).
Defn: To gain or regain possession of something in exchange for a monetary payment.
3 Egomaniac insists car is restyled (10)
NARCISSIST : Anagram of(.. is restyled) INSISTS CAR.
4 Very pale, like domesticated fowl (5)
ASHEN : AS(like, as in “good as gold”) + HEN(female domesticated fowl).
5 Delight bear (9)
TRANSPORT : Double defn: 1st: To overwhelm with delight; and 2nd: To carry people or goods.
6 Camera-shy, concealing skin complaint (4)
RASH : Hidden in(…, concealing) “Camera-shy“.
7 Erratic mounted police contain riotous raid (8)
SPORADIC : Reversal of(mounted, in a down clue) COPS(informal term for policemen) containing(contain) anagram of(riotous) RAID.
8 Strangely agrees to accept the empty ship’s lowest class accommodation (8)
STEERAGE : Anagram of(Strangely) AGREES containing(to accept) “the” minus its inner letter(empty).

13 Idle, posh prisoner raised trophy bomb (10)
UNOCCUPIED : U(letter signifying “upper-class”/posh) + reversal of(… raised, in a down clue) CON(short for “convict”/a prisoner) + CUP(a sporting trophy) + IED(abbrev. for “improvised explosive device”/a simple bomb made and used by unofficial or unauthorised forces).
15 Tried out black vehicle with scarlet upholstery (9)
REHEARSED : HEARSE(a vehicle carrying a coffin, often, but not always, coloured black) contained in(with … upholstery) RED(scarlet being a brilliant red colour).

16 Wounding comment on notice over small island (8)
BARBADOS : BARB(a wounding comment/a deliberately hurtful remark) placed above(on, in a down clue) AD(abbrev. for “advertisement”/a promotional notice) + O(abbrev. for “over” in cricket scores) + S(abbrev. for “small”).
Defn: … in the Caribbean.
17 Transport union’s head replaced by Conservative, briefly (8)
CARRIAGE : 1st letter of(…’s head) “marriage”(a legally recognised union between two people) replaced by(replaced by) “C”(abbrev. for /briefly, a Conservative, a member of the Conservative Party
19 Emblem right for plague (6)
BADGER : BADGE(an emblem/a distinguishing object) + R(abbrev. for “right”).
Defn: To pester or harass someone continually.
20 Remove permit to enter river (6)
DELETE : LET(to permit/to allow) contained in(to enter) DEE(any one of a number of rivers in the UK and Australia).
23 Offspring lives with Susan (5)
ISSUE : IS(lives/exists) plus(with) SUE(shortened form of the name Susan).
Defn: … in legalese.
24 Heads of kingdoms negotiate international treaty and unite (4)
KNIT : 1st letters, respectively, of(Heads of) “kingdoms negotiate international treaty“.
Defn: …/to bond.
Very enjoyable. My favourites were 11a and 18a.
Thank you scchua and carpathian
Thanks Carpathian and scchua
Odd one for me. A lot of answers, particularly in the top half, were obvious enough after some crossing letters had gone in, but far from obvious from the clue by itself, which, I think, makes it on the hard side for a Quiptic.
I confess that I gave up and revealed REDEEM!
Thanks both, great stuff.
Re 12a, I know that F is often FINE abbreviated, but I can’t think why. Can anyone help?
Shirl @3
Chambers: Fine (on lead pencils)
ODE: British fine (used in describing grades of pencil lead)
Thanks Gaufrid. I only knew H & B on pencils!
Thanks Carpathian and scchua. I enjoyed this puzzle. The abbreviation OR for other ranks was new to me so LOI was BADGER. HEADSTONE, BARBADOS, & ANAESTHETIC gave me a smile when I came up with the answers after thinking in an entirely different direction. Wishing everyone a good week.
First and third letters of hoax for regularly, I think (ha).
Can someone help me understand 21 AC, please? It is probably blindingly obvious, however, I can not understand it. Appreciate any help. Otherwise a great crossword, with great explanations.
Martin, It’s simply the letters B-A-R-E hidden, in reverse, inside the word “vulnerable”. So “exposed” is the definition of bare.
Many thanks, davlo, must have had a senior moment. Again many thanks.
Did no one but me try to fit RCMP into 7 down?