Across:
1 What one exercising strong-arm tactics produces perhaps (BICEPS) in seafood (mussels) report (= homophone indicator) (‘mussels’ = ‘muscles’ = BICEPS),
4 Fellow (ACADEMIC) produced a (A-) computer-aided design (computer-aided design = -CAD-) to train (= anagram indicator) mice (mice = -EMIC) (ACADEMIC),
9 Quick movement (NIMBLE) of 1000 (Roman 1000 = -M-) bass (-B-) in river (River Nile = NI-LE) (NIMBLE),
10 It’s revealed (-OUT) after (= position indicator) strike (KNOCK-) (KNOCKOUT) – it’s the final blow (KNOCKOUT),
12 At one’s wits end (IN DESPAIR), dines (dines = IN DES-) out (= anagram indicator) with Castor and Pollux (-PAIR) (IN DESPAIR),
13 The New Yorker’s (the New Yorkers = YANKS) local readership in Washington, for instance (YANKS),
14 Be given to understand (HEAR-) initial trouble (initial ‘Trouble’ = -T) and lack of success (FAILURE) (HEART FAILURE) could be the death of you (HEART FAILURE),
18 Paid for opportunities for further study (SCHOLARSHIPS) in 4 across similar (‘academic’ is similar to SCHOLAR-) to those produced in 6 down (-SHIPS produced in ‘dockyards’) (SCHOLARSHIPS),
21 No tin in salt mine (no ‘tin’ in ‘salt mine’ = salme = MEALS) for the consumers (MEALS),
22 Join up (fuse = F- -USE) around lough (Lough -REE) to draw attention (HO-) to (FREE HOUSE) party that appeals to students (FREE HOUSE),
24 Drink (sup = SU- P-) around back of saloon (back of ‘salooN’ = -N) or (-OR-) church (-CH) (SUN PORCH) in room getting lots of light (SUN PORCH),
25 Sailed (sailed) away (= anagram indicator) (sailed = LADIES) with the mates perhaps (LADIES),
26 Inadequate (DERISORY) single (-I-) in New (= anagram indicator) Order’s (orders = DER-SOR-) last anthology (last ‘anthologY’ = -Y) (DERISORY),
27 In the old sense, veteran (old STAGER) runs (-R) behind (= position indicator) playground? (play ground = STAGE-) (STAGER).
Down:
1 Presumably, this is how Johnston, Mooney and O’Brien settle their disputes (BUNFIGHT) for The Tea Party? (BUNFIGHT),
2 A demonic (a demonic) boss (= anagram indicator) (a demonic = COMEDIAN) is acting the eejit for the audience (COMEDIAN),
3 Extrapolates dropping tax rate (‘extrapolates’ dropping ‘tax rate’ = poles = POLES) for some Europeans (POLES),
5 Manage (CONTROL) Talking Heads (PANEL) (CONTROL PANEL) with set of instruments (CONTROL PANEL),
6 One of those in Court (DOCK-) exercising the criminals (-YARD-) is taking direction (south = -S) (DOCKYARDS) from those fitting up crafty types (DOCKYARDS),
7 What those around Jersey (cattle) are hearing (MOOING cattle) is coming from Dexter (cattle), in a manner of speaking (MOOING cattle),
8 The cleverest of them all (CUTEST) from college (C-) and university (-U-) on trial (-TEST) (CUTEST),
11 One working overseas (NAVAL OFFICER) for Orange (‘navel’ orange) State (= homophone indicator) (‘navel’ = ‘naval’ = NAVAL) in a mess (OFFICER) (NAVAL OFFICER),
15 Lost the head (s) in a tight spot (stuck) (‘stuck’ without ‘s’ = TUCK) and informs on local (‘informs’ on locally = SHOPS) (TUCK SHOPS) selling to students from a number of schools (TUCK SHOPS),
16 Luxembourg (-LU-) metal (-TIN-) found in quarry (DI-G) (DILUTING) is weakening the mix (DILUTING),
17 Jenny (ASS-) is key (music key -E-) to backing Ross (-SSOR-) (ASSESSOR) in The Examiner (ASSESSOR),
19 Occupied (AMUSED) by a (A-) thought (-MUSED) (AMUSED),
20 A certain approach (MANNER) from Orangemen with no ego (‘orangemen’ with no ‘ego’ = ranmen = MANNER),
23 Opens 14 across (opens ‘heart failure’ = HEART) with enthusiasm (HEART).