Across:
1 Recognition (HOMAGE) for drawing attention (HO-) to wild (= anagram indicator) game (game = -MAGE) (HOMAGE),
4 Trim CIA’s (trim CIA’s = SCIMITAR) organisation (= anagram indicator) for crossing swords in the East (SCIMITAR),
9 Not removed from rationale (‘not’ removed from ‘rationale’ = raiale = AERIAL) that gives you a better picture (AERIAL),
10 Analyse the vulnerable (PSYCH OUT), pathetic (= anagram indicator) spy (spy = PSY-) caught (-C-) by Bellow (shout) losing the head (s) (‘shout’ without ‘s’ = -H OUT) (PSYCH OUT),
12 Built poor (built poor = POLITBURO) structure (= anagram indicator) for The Party’s Committee (POLITBURO),
13 Live (DWELL) with director (D-) harmoniously (-WELL) (DWELL),
14 Keep an eye on (MIND -) swamp (-BOG-) with great (-G-) fish (-LING) (MIND-BOGGLING) – it’s astonishing (MIND-BOGGLING),
18 A small word (ABBREVIATION) with one of those in USA, literally speaking (one of “U or S or A” is an ABBREVIATION),
21 Sounds like (= homophone indicator) a film (a film = “a veil” = “avail” = AVAIL) to take advantage of perhaps (AVAIL),
22 Properly manage (HANDLE-) the pub (-BAR) (HANDLEBAR) or on your bike! (HANDLEBAR),
24 Aware (CAUTIOUS) of accountant (CA-) from Utah (-UT-) with debts (IOUs = -IOUS) (CAUTIOUS),
25 Left (L-) Adam (Adam = -AM-DA) Cook (= anagram indicator) over open bigotry (open ‘bigotry’ = -B-) (LAMBDA) in Greek letter (LAMBDA),
26 Leaves home perhaps (tea leaves home in a TEA CHEST) by train (TEA CH-) with set (set = -EST) plan (= anagram indicator) (TEA CHEST),
27 Manchester has no hams (‘Manchester’ has no ‘hams’ = nceter = CENTRE) or heart (CENTRE).
Down:
1 Pressure from local Garda (HEAT) Force (force = PUMP) (HEAT PUMP) is bound to raise the temperature (HEAT PUMP),
2 March (MAR-) on Fourth of July (fourth of ‘July’ = -Y) on open space (-LAND) (MARYLAND) in American state (MARYLAND),
3 Shoot (shoot/bud = GRAFT) lots of birds (-RAFT of birds) after (= position indicator) midnight (mid ‘night’ = G-) (GRAFT),
5 Settling existing accounts (COST OF LIVING) – sounds like something pricey the economists came up with (COST OF LIVING),
6 A comedian (a comedian = MACEDONIA) working (= anagram indicator) in The Balkans (MACEDONIA),
7 Starting trouble (starting ‘trouble’ = T-) and argument (-ROW-) with fan (fan belt) at centre (“belt” at the centre = -EL) (TROWEL) – it’s what one of those on site has to handle (TROWEL),
8 New (= anagram indicator) alert (alert = RAT-LE) over initial trouble (initial ‘trouble’ = -T-) (RATTLE) with nursery toy (RATTLE),
11 Flow (SURGE-) is never-ending (‘never’ ending = -R-) from artery, ultimately (‘artery’ ultimately = -Y) for periods over the day (HOURS) (SURGERY HOURS) – it’s time to see a doctor (SURGERY HOURS),
15 Duck (BOB-) from butcher? (butcher = “slay” = “sleigh” = -SLEIGH) That sounds like (= homophone indicator) (BOBSLEIGH) something you might get for a snowy Christmas! (BOBSLEIGH),
16 It’s part of the drill (drill BIT) opening bloodstock (opening ‘bloodstock’ = B-) yard (= -Y) – that’s from the horse’s mouth (BIT) (BIT BY BIT), gradually! (BIT BY BIT),
17 Mount up (INCREASE) for a sincere (a sincere = INCREASE) groom (= anagram indicator),
19 Powerful businessman (FAT CAT) from Greece (“Greece” = “grease” = FAT) told (= homophone indicator) to let it (cat) out of the bag (‘let the CAT out of the bag) (FAT CAT),
20 The foreign (the = LA-) copper (-CU-) leads new (-N-) agency (-A) (LACUNA) – it concerns gap in statement perhaps (LACUNA),
23 Longs to get out of Los Angeles (‘longs’ to get out of ‘Los Angeles’ = aeles = LEASE) contract for Homeland, perhaps (LEASE).