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ACROSS

 1 Cut the budget severely to pinch a penny—that’ll make a big impression (6)

 4 Where dozens can be found with wounds or bruises (8)

 9 Cause trouble when discussing something that helped Charles get around?
  (5,4)

11 Wear down the rodent’s insides (5)

12 Tiny measure of anxiety from getting beheaded (8)

13 Space object sounds more substantial (6)

15 Host ignoring the odds of team scheme (5)

17 Precious walk-around, indeed (9)

18 French guardsman’s projectile: silver bread (9)

20 Take place of Orange County dog (5)

22 Obnoxious kid from Southern California is the subject of experiments (3,3)

23 See a friend beginning to mount a fruit tree (4,4)

26 Letter from the professor’s helper (5)

27 Fire engulfs center of Odessa neighborhood belonging to a bygone emperor
  (9)

28 Guard dispatched in a train (8)

29 Where to keep a record of mischievous elves: Endor, at first (6)

DOWN

 1 Outward appearance of an expert on the board? (7)

 2 Telling falsehoods in bed? (5)

 3 Beloved soprano with small fashion accessory (7)

 5 Grandma picking up two articles (4)

 6 Vain American embraces tennis champion (7)

 7 I went over a little gimmick, without finishing a type of exercise (9)

 8 Heartless Eve, in ragged dress, gets mad (4,3)

10 Taking on giant alien holding up shell (8)

14 Inexperienced top to bottom? Time to hire again (8)

16 West Indian pasta, for example, including one legume (9)

18 Weakens insult about a stringed instrument (7)

19 In Florida, NASA zigzags with ancient inhabitants of the Southwest (7)

20 Shockingly, Rove calls naked general (7)

21 Escalating charm sustains average love affair (7)

24 British pop diva’s first-class strike (5)

25 Starts to annotate clinical history: “Experiencing pain” (4)

SOLUTION TO PUZZLE NO. 3376

ACROSS 1 SAC[k] + AJA[r] + WEA[r] 6 A + SAP 10 LOO(KING + A)T 11 [r]OSIER 12 “shy, Anne” 13 anag. 15 “mop it” 17 EMPO (anag.) + RIA (rev.) 19 hidden 20 E(NCLA)VE (clan anag.) 22 Z(APP)ED 25 [s]-EXIST-S 26 AR(GU + MEN)T 29 hidden 30 TWOS + EATER 31 [t]EDDY 32 BODY(P)ARTS (bad story anag.)

DOWN 1 S-ELECT 2 anag. 3 JU + ICE 4 WA(GONE)TT + E + S 5 [f]ACTS 7 S + HIPPER 8 PAR + MESAN (anag.) 9 LOLL(I)POP 14 GEE + ZERHOOD (anag.) 16 anag. 18 RELE (anag.) + GATE 21 CRIB + BED 23 ERE + CTOR (rev.) 24 UT URNS 27 US + EU + P 28 S + TAB