Across:
1 Cheesy types (WHEELS of cheese) of local cars (“local’ cars = WHEELS),
4 Promoted (ADVANCED) Dan Cave (Dan Cave = ADVANCE-) ridiculously (= anagram indicator) to director (-D) (ADVANCED),
9 Bill (NOTICE) and Max deducted from income tax (“Max” deducted from “income tax” = incoet = NOTICE),
10 They hold the residues from 25 across (ASHTRAYS hold the residues of “cigars”) for a shy arts (a shy arts = ASHTRAYS) model (= anagram indicator),
12 Mystify (PERPLEX-) writer (-ED) (PERPLEXED) at sea (PERPLEXED),
13 Local trouble (AGGRO) with a (A-) horse (-GG-) half way down the road (half of “road” = -RO) (AGGRO),
14 Group of performers (CAST-) consumed by Band Aid’s (Band Aids = PLASTER -S) (PLASTER CASTS) material for those getting their big breaks perhaps (PLASTER CASTS),
18 Get the wrong end of the stick (MISINTERPRET), mostly (s) addressing teacher (Miss) (“miss” without “s” = MIS-) to explain the meaning (-INTERPRET) (MISINTERPRET),
21 Acknowledge (THANK) what’s receding out of kitchen garden (“receding” out of “kitchen garden” = kthan = THANK),
22 Duff (= anagram indicator) rang Given (rang Given = ENGRAVING) for making an impression on the Cup (ENGRAVING),
24 Restrict the consumption of (R-ATION) good books (Old Testament = -OT-) inside (ROTATION) revolution (ROTATION),
25 Hop off cargo ship (“hop” off/from “cargo ship” = cargsi = CIGARS) coming out of Cuba (CIGARS),
26 Position batsman takes perhaps (cricket batsman facing bowler is “on strike” = ON STRIKE) is not working (ON STRIKE),
27 Get lost (BEAT IT) in Tibet? A (Tibet a = BEAT IT) mistake! (= anagram indicator).
Down:
1 Capable of giving air to a character (WINDPIPE) like Twist (WIND-) in Great Expectations (-PIP-) from the first edition (first edition = -E) (WINDPIPE),
2 On the outside (EXTERNAL), this refers to The Examiner (the EXTERNAL examiner),
3 Sounds almost like (= homophone indicator) it could be of interest to someone on a diet (“low cal” = “lo cal” = LOCAL) in the neighbourhood (LOCAL),
5 Unfinished (c) record (‘disc’ without ‘c’ = DIS-) by The Edge (-ADVANTAGE) (DISADVANTAGE) is a stumbling block (DISADVANTAGE),
6 A (A-) retweet (-RT) by dog’s owner (MASTER) (ART MASTER) is the fastest on the draw presumably (ART MASTER),
7 Credit (CR-) a (-A-) horse (-GG-) from yard (-Y) (CRAGGY) that’s weather-beaten (CRAGGY),
8 Posted (posted = DESPOT) different letters (= anagram indicator) to ruthless leader (DESPOT),
11 Classy record (EXERCISE BOOK) of what one studied when working out (EXERCISE BOOK),
15 Restrict (TRIC-STER) sport (= anagram indicator) over the weekend (week end = -K-) (TRICKSTER) synonymous with April Fool’s Day (TRICKSTER),
16 Institute (I-) writing (from 3R’s = -R-) with bluster (-R-ANT) over city centre (“city” centre = -IT-) (IRRITANT) will only inflame things (IRRITANT),
17 Moving (= anagram indicator) gates (gates = STAGE) on park (SET) (STAGE SET) to play area (STAGE SET),
19 Initial storm (initial “storm” = S-) damages (= anagram indicator) tree (tree = -TERE-) hooked up to old (-O) (STEREO) sound system (STEREO),
20 Women (W-) removing the top (t) samples (“tastes” without “t” = -ASTES) (WASTES) from drains (WASTES),
23 Local chipper (local/informal “chipper”/lively = ALIVE) gets a (A-) vile (vile = evil = -LIVE) review (= reversal indicator) (ALIVE).