Thanks Raich for a quick enjoyable solve. Definitions are underlined in the clues.
Across
1 Spin doctors inaccurate about Society writer (6)
PROUST : PR(abbrev. for “Public Relations”, a term to refer to spin doctors;spokespersons who give a positive spin to events) + OUT(inaccurate, as in “his calculation was out by a few thousand”) containing(about) S(abbrev. for “Society”).
Answer: Marcel, French writer, whose most prominent work was translated as “Remembrance Of Things Past”.
4 Entertainment, retrospective, from latter epoch (8)
OPERETTA : Reversal of(retrospective) and hidden in(from) “latter epoch“.
9 Obsessive about the writer’s dog maybe (6)
ANIMAL : ANAL(obsessive;short for anal-retentive;excessively orderly and fussy) containing(about) I’M(writer’s;writer is;I am, used self-referentially).
Defn: …;for example.
10 Potentially solitary 16-21 (8)
ROYALIST : Anagram of(Potentially) SOLITARY.
Defn: Anti(answer to 16 across)-Parliamentarian(answer to 21 across), a supporter of King Charles I in the English Civil War.

11 Knack associated with born goalie in play (4,4,2,5)
LOOK BACK IN ANGER : Anagram of(… in play) [ KNACK plus(associated with) BORN GOALIE] – requiring “play” to do double duty.
Alternatively: Anagram of(… associated with …) [KNACK + BORN GOALIE] – a not-so-explicit anagram indicator.
Defn: … by John Osborne.
13 Possibly recollecting lines no longer used such as former astronomical systems (10)
GEOCENTRIC : Anagram of(Possibly) [ “recollecting” minus(… no longer used) “l,l”(abbrev. for “line” x 2).
Defn: Describing the former Ptolemaic astronomical model which put Earth as the centre of the universe.
14 Shabby poet recalled (4)
DRAB : Reversal of(… recalled) BARD(a poet).
16 Worker one finds hostile (4)
ANTI : ANT(an insect who could be one of the worker caste in its social system) + I(Roman numeral for “one”).
18 Ensure no more is heard of girl I abhor, hugging the Italian? (10)
ANNIHILATE : ANN(a girl’s name) + I + HATE(to abhor;to extremely dislike) containing(hugging) IL(“the” in Italian).
21 Roundhead, Martin, a real pain possibly (15)
PARLIAMENTARIAN : Anagram of(… possibly) MARTIN, A REAL PAIN.
Defn: One who fought against King Charles I in the English Civil War.

23 Get rid of drink ensnaring a 21 (5,3)
STAMP OUT : STOUT(a strong, dark beer) containing(ensnaring) [A + MP(abbrev. for a Member of Parliament;answer to 21 across) ].
24 Rope is in possession of queen, wife’s admitted (6)
HAWSER : HAS(in possession of) ER(abbrev. for Elizabeth Regina, the Queen) containing(…’s admitted) W(abbrev. for “wife”).
25 One getting what he wants, embracing Church, excellent example (8)
SCORCHER : SCORER(one who succeeds;gets what he wants, often in reference to a sexual conquest) containing(embracing) CH(abbrev. for “church”).
Defn: An excellent or remarkable example of something, such as a pretty girl or an absorbing novel.
26 Outsiders in race, choke, dawdle, become more distant (6)
RECEDE : 1st and last letters, respectively, of(Outsiders in) “race, choke, dawdle“.
Down
1 Place holding attention – one much valued (5)
PEARL : PL(abbrev. for “Place”) containing(holding) EAR(attention, as in “to give one your ear”).
2 Operations Research used by popular old firm – flower of America (7)
ORINOCO : OR(abbrev. for “Operations Research”) plus(used by) IN(popular;in fashion) + O(abbrev. for “old”) + CO(abbrev. for “company”;a commercial firm).
Defn: A flow-er;river in South America.
3 Fake blessing – not half a mess! (8)
SHAMBLES : SHAM(a fake;a fraud) + “blessing” minus its last 4 letters(not half).
5 It might herald a long dry period? (11)
PROHIBITION : Cryptic defn: Reference to the period 1920 – 1933 in the US when the manufacture, transportation and sale of alcoholic drinks were prohibited;a long period to be alcohol-free;dry. Needless to say, quite a few broke the law.
6 Ronald referring to silver article (6)
REAGAN : RE(refering to;with reference to) + AG(symbol for the element, silver) + AN(an article in grammar).
Defn: Surname of the late US President with the given name, Ronald.
7 Set off briefly describing branch of mathematics, German (7)
TRIGGER : TRIG(short for;briefly describing “trigonometry”, a branch of mathematics) + GER(abbrev. for “German”).
8 A tall beer, dreadful, can be changed (9)
ALTERABLE : Anagram of(…, dreadful) A TALL BEER.
12 Cunning manoeuvres from Tom and Jerry? (3-3-5)
CAT-AND-MOUSE : Cryptic defn: Reference to the cartoon characters, Tom, a cat, and Jerry, a mouse.

13 They hold up bar (9)
GOALPOSTS : Cryptic defn: Reference to the crossbar held up by 2 posts in, say, football.
15 Bad habit, pinching a newspaper where clergyman lives (8)
VICARAGE : VICE(a bad habit) containing(pinching) [ A + RAG(a somewhat derogatory term for a newspaper).
Defn: A vicar’s home.
17 Twister in two minds over fuss (7)
TORNADO : TORN(in two minds;undecided) placed above(over, in a down clue) ADO(a fuss;a commotion).
19 America at first supported by paper in disagreement (2,5)
AT ISSUE : 1st letter of(… at first) “America” placed above(supported by, in a down clue) TISSUE(a disposable piece of absorbent paper).
20 Film couple describing work in long-gone era (6)
BIOPIC : [ II(Roman numeral for “two”;a couple) containing(describing) OP(abbrev. for “opus”;a musical or literary piece of work) ] contained in(in) BC(abbrev. for “Before Christ”, denoting the years before the birth of Christ;a long-gone era).
Defn: … relating the life story of a real person.
22 Perhaps tend injured shark (5)
NURSE : Double defn.

Nice, neat, simple. QED (Quite Easily Done!)
What Nicola said!
Thanks again for going the extra mile, Scchua. Being in my second childhood, I got a kick out of extrapolating the Tom and Jerry scenario.
Thank you, Scchua, for the excellent blog with the fine illustrations. Thanks also to those who commented.