Across:
1 What you might do if you haven’t got Hoover (SWEEP) directors (THE BOARD) (SWEEP THE BOARD) to win everything (SWEEP THE BOARD),
10 Brave (WARRIOR) reading (from the 3Rs = -R-) and writing (from the 3Rs = -R-) in Iowa (Iowa = WA-IO-) travelling (= anagram indicator) (Iowa = WA-IO-) by river (-R) (WARRIOR),
11 Introduces a suitable girl (introduces ‘A Suitable Girl’ = -ASG-) in scarlet (glow = GL-OW) (GLASGOW) to The City, in Britain (GLASGOW),
12 Not best pleased (sore) upsetting (= reversal indicator) (sore = EROS) lover (EROS),
13 What the last defender (sweeper) might do (SWEEP) on all sides today (SWEEP on all sides of today’s grid),
15 A white alkaline compound (LIME) concealed by the military (concealed by ‘thE MILitary’ = emil) turning up (= reverse indicator) (emil = LIME),
17 Couldn’t get rid of cold (‘couldnt’ get rid of ‘cold’ = unt = NUT) in Brazil (NUT),
19 One (I-) doing (doing = -NDIGO) work (= anagram indicator) (INDIGO) out of the blue? (INDIGO),
21 Hold (hold = OLD H-) off (= anagram indicator) getting a (-A-) shirt (-T shirt) (OLD HAT) that’s no longer fashionable (OLD HAT),
22 Remove head (c) from canna (canna) (‘canna’ without ‘c’ = ANNA-) and head from tulip (head from ‘Tulip’ = -T-) to (-TO) (ANNATTO) use in food colouring (ANNATTO),
23 Soldier (GI = G-I-) admits not being in good shape (-ILL-) taking ecstasy (-E) (GILLIE) working for a sporty type where there’s a catch (handle the catch/fish = GILLIE),
25 What’s the point (point in fencing = TOUCHÉ) in saying you got me there clever clogs! (acknowledge a valid point/clever remark ‘touché’ = TOUCHE)
27 It’s central (MID) to what’s covered up by brand imagery (what’s covered up by ‘branD IMagery’ = dim) going all the way back (= reverse indicator) (dim = MID),
29 Living life in the fast lane at the opening and closing (‘LIving life in the fast laNE’ at the opening and closing = LINE) of bar (LINE),
30 Hawaiian parties (LUAUS) in light rail system (LUA-S) going around university (university = -U-) (LUAUS),
31 Reveal a policy covering up (‘reveaL A POlicy’ covering up = lapo) review (= reversal indicator) (lapo = OPAL) by one of The Stones (OPAL),
34 Turns up (= reverse indicator), a measure of fat content (Body Mass Index = BMI = IMB-) is (is = -I-S) found around heads of blackbirds, eiders (heads of ‘Blackbirds Eiders’ = -BE-) (IMBIBES) and swallows (IMBIBES),
35 It’s sprinkled over pizza (OREGANO) in Genoa or (Genoa or) all over the place (= anagram indicator) (Genoa or = OREGANO),
36 Breaking down (-WEEPING) profit (GAIN-) in ship (SS = S- -S) (SWEEPING GAINS) producing comprehensively more (SWEEPING GAINS).
Down:
2 Running low on tips (‘Running Low On’ tips = -RLO-) in administrative district (ward = WA-RD) (WARLORD) for aggressive regional leader (WARLORD),
3 Get Naomi to leave mediation (get ‘Naomi’ to leave ‘mediation’ = edti = EDIT) or change one’s story? (EDIT),
4 The (the) endless (e) (‘the’ without ‘e’ = TH-) trouble (-ROW-) with new (-N) (THROWN) cast (THROWN),
5 Legate (legate) travelling (= anagram indicator) (legate = EAGLET) with young high-flyer (EAGLET),
6 Dirge from Gore Vidal (‘dirge’ from ‘Gore Vidal’ = oval = OVAL) in a roundish way relates to a presidential office (OVAL Office),
7 Ben from Neighbours (‘Ben’ from ‘neighbours’ = ighours = ROGUISH) is the mischievous type (ROGUISH),
8 Raising dust (SWEEPING) on request (CLAIM) (SWEEPING CLAIM) or is that a generalisation? (SWEEPING CLAIM),
9 Clean up (SWEEP THE FLOOR) wet sheep (wet sheep) neglected (= anagram indicator) (wet sheep = SWEEP THE) by 18 down (‘floor’ = FLOOR) (SWEEP THE FLOOR),
14 University (-U-) in dodgy (= anagram indicator) area is (area is = E-RASIA) (EURASIA) offering a multitude of places all the way to China (EURASIA),
16 Part of Malaga invokes (part of ‘malAGA INvokes’ = AGAIN) another time (AGAIN),
18 Stun (FLOOR) 50 (Roman 50 = -L-) in The Upper House (roof = F-OOR) retiring (= reverse indicator) (FLOOR),
20 Neo- (neo) -liberal (- anagram indicator) (neo = ONE) ace (ace = ONE),
21 It’s served up (= reverse indicator) in Trattoria (in ‘traTTOria’ = tto = OTT) or is that an exaggeration initially? (‘over the top’ = OTT),
24 Depressed (low = L-OW) over broadcast (-ON-) from European island (Great Britain = -GB-) – the first one (the first ‘One’ = -O-) (LONGBOW) picked up by one of The Archers (LONGBOW),
26 Leader (CAPTAIN) of school committee (parent-teacher association = -PTA-) taken in by murderer (Cain), the one that wasn’t able (wasn’t able) reportedly (= homophone indicator) (wasn’t ‘able’ = wasn’t ‘Abel’ = Cain = CA-IN) (CAPTAIN),
27 Manhattan upper east side’s starters (‘Manhattan Upper East Sides’ starters = MUES-) on half the menu (list) (half of ‘LIst’ = -LI) (MUESLI) for breakfast (MUESLI),
28 Used a shovel (DUG-) to find unpleasant smell (pong) beginning to go off (p) (‘pong’ without ‘p’ = -ONG) (DUGONG) produced by sea cow (DUGONG),
32 Music producer (OBOE) consumed by video box set (consumed by ‘vidEO BOxset’ = eobo) backtrack (= reverse indicator) (eobo = OBOE),
33 First mate cutting Trim’s (‘first mate’ cutting ‘Trims’ = fate = FETA) goat’s cheese (FETA).