3985: Yes by Lato

Lato presented us with a sort of theme and variations puzzle. Eleven entries were associated, one incompletely, with various members of a group. These were  defined in their full associated versions by one extra word in each of eleven clues. The title referred to the once popular TV series Yes Prime Minister  and the group was the last 12  British prime ministers. Three answers gave this message – DEPRESSING PLACE IN SOMERSET, a cryptic reference to Downing Street. The remaining PM was Sir Anthony Eden and the EDEN PROJECT completed the links. Solvers had to write the seven-letter word PROJECT below the grid to complete the puzzle.

In clue order the associations are as follows.
Maggie (Thatcher) MAY [song]
Jim (Callaghan) CRACK [worthless]
SMART Alec (Douglas-Home) [know-all]
Brown BETTY [pudding]
Macmillan NURSES [tenders]
Blair WITCH (PROJECT) [film]
Major MITCHELL [cockatee]
Clement- (Attlee) INES [fruit]
Wilson PICKETT [singer]
Heath HENS [grouse]
ROBERT Winston (Churchill) [professor]


Extra words in clues are shown in square brackets.
ACROSS
1 Preparing covering letter (10)
DEPRESSING (wordplay only) EP (Epistle) in DRESSING
9 Drive round in wreck (4)
RUIN I in RUN
10 Bullock in drama involving men out of southern country (7)
ANDORRA SANDRA (Bullock) less S containing OR (men = other ranks)
12 Told to provoke small dog (4)
PEKE homophone PIQUE
13 Drop talking [cockatoo] causing row (4)
TIER Homophone TEAR
15 Priest born in Belgian city back paying homage (8)
KNEELING ELI (priest) + N (born) in GENK (reversed)
18 John, perhaps, boxing in second bout (4)
DOSE S in DOE (ref. John Doe)
22 He stops tide turning, giving praise to God (5)
PAEAN A (he) in NEAP (reversed)
24 Saucy broad gets a thousand for story at first (6)
CHEEKY CHEESY (broad, as of smile) K for S
25 Sandy's mount has character - not worried (5)
SCLIM (climb in Scots) S (has) + CLIM(ate) less ATE (worried)
26 Triumphant exclamation about Italy - Columbus's [film] location (4)
OHIO I (Italy) in OHO
29 [Worthless] fool eating master's porridge (4)
SAMP M in SAP
31 Pass over bad [singer] (4)
EVIL LIVE reversed
34 Salt to sprinkle is on edge of table (5)
EOSIN anagram IS ON + E (last letter of TABLE)
35 Soft, delicate material (5)
PLACE (wordplay only) P + LACE
36 [Professor] informs on doctor breaking laws (7)
RUMBLES MB in RULES
37 Prove right [know-all], missing nothing (3)
TRY TORY less O
38 Engineer met seniors (10, two words)
IN SOMERSET (wordplay only) anagram MET SENIORS
DOWN
1 Doctor of Philosophy's first lapse (4)
DROP DR + O (of) + P(hilosophy)
2 O'Neill deprived of two points - well done (4)
EUGE EUGENE less NE (2 points) (Ref. Eugene O’Neill)
3 Will's likewise largely overturned - not right (6)
EGALLY anagram LARGELY less R
4 Prepared to accept contractor's initial [tenders] for school (4)
SECT C in SET
5 Tearing off artificial skin (7)
INTEGRA anagram TEARING
6 Flipping dog biting mother might make you this? (5, two words)
GO MAD MA in DOG (reversed)
7 A pair of rooks land - one comes in for earth, one part of fence (9, two words)
ARRIS RAIL A + RR (2 rooks) + ISRAEL (I for E (earth))
8 Incessant talk about the river rising (6)
YATTER RE + T (the) + TAY (all reversed)
10 One endlessly important piece of nature (4)
AKIN A (one) + KIN(G)
11 Stirs drink served up (4)
ADOS SODA (reversed)
14 Berry [song] has Prince on piano (4)
RASP RAS (prince) + P
16 Be near daughter leaving old Edinburgh ground (9)
NEIGHBOUR anagram O (old) + EDINBURGH less D (daughter)
17 Eat [pudding] away from headless apparition (4)
ETCH (F)ETCH
19 No depth to darkish river (3)
USK DUSK less D
20 Get time off to tutor every one separately (4)
EACH (T)EACH
21 Kelvin wearing upturned bowler in new versions (7)
REMAKES K (Kelvin) in SEAMER reversed
23 Quits taking the French teams (7)
ELEVENS LE (the in French) in EVENS
24 [Grouse] about Italian's sons - not gentlemen (4)
CITS C (about) + IT + S
25 Abandon earl in fix (6)
SCRAPE SCRAP + E (earl)
27 African animal raised in pen (5)
ORIBI I (in) + BIRO (reversed)
28 With priest exorcised solicitor's demons (4)
IMPS PIMPS less P (priest)
30 Reddish purple swelling soft on top (4)
PLUM LUMP with P at start
32 French thought mostly the same about Spain (4)
IDÉE IDE(M) + E (Spain)
33 [Fruit] most unlikely to remain fresh (4)
LAST 2 definitions