Across:
1 Strips (peels = SLEEP-), retiring (= reversal indicator) and closing eyes (closing/last letter “eyes” = -S) (SLEEPS), drops off (SLEEPS),
4 Narrate (narrate = RENT-A- -AR) novel (= anagram indicator) about head case (head “case” = -C-) (RENT-A-CAR) leaving the airport (RENT-A-CAR),
9 Madeleine’s dropping aims (“Madeleines” dropping “aims” = delene = NEEDLE) that will annoy someone (NEEDLE),
10 One of those onto the Letter’s (STAMP-) Editor (-ED-) is engineering (-E) (STAMPEDE) panic (STAMPEDE),
12 A long rambling story (RIGMAROLE) of outfit (RIG-) with secret agent (mole = -M-OLE) capturing a (-A-) ringleader? (“ring” leader = -R-) (RIGMAROLE),
13 The sense (one of the senses = TASTE) to compliment a stewardess inside (“compliment a stewardess” inside = TASTE),
14 Enthusiasm (PASSION) for Apple (FRUIT) (PASSION FRUIT) grows in tropical areas (PASSION FRUIT),
18 We owe it to ourselves perhaps (NATIONAL DEBT) to state (NATIONAL) cause of bankruptcy (DEBT) (NATIONAL DEBT),
21 Cover story (ALIBI) in Bengali biweekly (in “Bengali biweekly” = ALIBI),
22 The Taoiseach’s buddy perhaps (The HOUSE MATE) is sharing with the one who can’t get deposit from parents for his own home (HOUSEMATE),
24 Pick up where one left off (CONTINUE) in résumé? (résumé = resume = CONTINUE),
25 Cream (BAL-M) covers tops of strawberries and (tops of “strawberries and” = -SA-) (BALSAM) something that’s naturally fragrant and soothing (BALSAM),
26 Detectives (detectives = DIs = DIS-) work out (-AGREE) (DISAGREE) how to cross Swords (cross swords = DISAGREE),
27 Flighty type (GLIDER) upsets (= reversal indicator) revolutionary (Red = -DER) in local government, for starters (“in local government” for starters = ILG = GLI-) (GLIDER).
Down:
1 They have a reputation for exposing people to the heat (SUNTRAPS) from newspaper (SUN-) stings perhaps (-TRAPS) (SUNTRAPS),
2 Greene is (Greene is = ENERGISE) stupid (= anagram indicator) to get a shot in the arm (ENERGISE),
3 Crew missing from workplace (“crew” missing from “workplace” = okpla = POLKA) might be seen at the dance (POLKA),
5 Nice businesswoman (Nice/French “businesswoman” = ENTREPRENEUR) from The Hustler? (ENTREPRENEUR),
6 What the judge gave (TIME) in court (TRIAL) (TIME TRIAL) challenges the biker (TIME TRIAL),
7 Posers would say (CHEESE) cheerleaders empty larder (“cheerleaders” empty “larder” = cheees = CHEESE),
8 The man in Ring (man in boxing ring = referee = RE-F-) keeps key (music key = -E-) that opens entrance rattling (opens “entrance rattling” = -ER) (REEFER) in jacket (REEFER jacket),
11 City Hall perhaps (COUNCIL HOUSE) is built on estate (COUNCIL HOUSE),
15 Supposing (IMAGINING) Parisian friend (Parisian/French “friend” = ami = IMA-) serves up (= reversal indicator) something to drink (-GIN-) in (-IN-) first glass (first “glass” = -G) (IMAGINING),
16 Revolutionary (RE-D) admits state (= homophone indicator) revolutions (“laps” = “lapse” = -LAPSE-) (RELAPSED) reverted to type (RELAPSED),
17 Are terms (are terms = STREAMER) arranged (= anagram indicator) for the party? (STREAMER),
19 Had a ball perhaps (DANCED) in The Man (“Dan the man” = DAN-) from Clare (-CE-) with leading director (leading “director” = -D) (DANCED),
20 Diabolical (= anagram indicator) pain (pain = PIAN-) is so (so = -OS) upsetting (= reversal indicator) (PIANOS) for the musicians (PIANOS),
23 Some Jerusalem ailments (some “Jerusalem ailments” = EMAIL) read online (EMAIL).