This puzzle seemed, frankly, a little underwhelming compared  to Mudd’s usual standard. Not that there is anything wrong with it, per se; just seems a tad light on imagination or inspiration. I suppose that’s the down side of publishing setters’ identities - it can start to frame certain expectations in the longer run.

Across
1 DESERT - double def’n.  I wasted a minute or so here, looking for an answer ending in sere (”dry”).
4 DRESSING - A “saucy” surface reading, if you’ll pardon the pun.
10 SEA(U)RCHIN[-g]
12 REED - Deer(rev)
13 SEWAGE FARM - E(nergy) in (a few grams)*
15 TRAIPSE - (pirate)*
16 KA,RATE - I’m willing to indulge “chopping” as the definition here, but the leeway is only because it clearly works best with the intended surface.
19 S,TENCH
21 H(ADD)OCK - “reckon”=add, as in a ready-reckoner.
23 PIL,LOW,CASE - Lip(rev), “blue” = mopey/low, “instance”=case (of).
25 CHIN[-a] - put me in mind of the old (and probably politically incorrect) barb; “He’s got more chins than a Chinese telephone directory”.
27 EG,RE,T
28 ANALOGISE (lose again)*
29 PE(AGREE)N
30 COCK UP - double def’n. As in the catchphrase of Geoffrey Palmer’s character in the Reggie Perrin series. “Bit of a cock up in the [.....] department”.

Down
1 DISCRETE - homophone of  “discreet”.
2 SPARE PART
3 RARE - double def’n.
5 RAN,SACK
6 SLAVE TRADE - (dealers + VAT)*
7 IN,DIA[-gram]
8 [-i]GLOO,MY
9 WHEEZE - double def’n.
14 SPIN DOCTOR - a clever idea, but perhaps too transparent?
17 TOO,TH(P)ICK - gets my vote for the clue of the day.
18 SKIN DEEP
20 HECTARE - (teacher)*
21 HUSS,A,R
22 UP,KEEP
24 LARVA - homophone of “lava: (which, I would pedantically point out, is only “hot stuff” in its molten state).
26 POLO - double def’n.

I forgot to solve this puzzle when I should have done, but when I recieved a “where’s that blog?” e-mail, decided I ought to be able to solve the puzzle, given that I (a) work for a dental software company and (b) have also been to the dentist lately.
I did so with a little help, mainly from the ‘ho’ in hihoba, and a bit more use of reference books than usual. Total time was maybe 4 hours.

The preamble told us that Nutmeg had been to the dentist recently, that Down clues had extra words whose first letters were three things used in treatment, and that we had to shade 14 letters in the grid to replace “that which is lost”.

13D was the clue that started me off on the thematic material, fairly obviously leading to LEXICOGRAPHER, three letters too long for the slot. The COG therein is of course a tooth on a wheel, so it looked very likely that some number of answers would lose teeth before entry. (The GAM in ADVENTUREGAME came out fairly easily too). This suggested some kind of DENTURE, though my initial guess of PERMANENT DENTURE was obviously too long. I had a P and E in the only two white squares in the grid (pink elsewhere), which seemed likely to be the start and end of the shading, so next guessed ‘PLASTIC DENTURE’. I wrongly tried first to fit the phrase as a ’smile’ shape, but when receiving my biggest chunk of help, mentioned my ? DENTURE idea, to which the response was “yes, now I can see it - PARTIAL DENTURE” (in a ‘tooth arch’ shape with start and end points as expected, going up to the ALDE in ALDERMEN). After more work to find the other teeth, we end up with:

Teeth (89 of them - one quadrant):

1A HIA(TUSH)ERNIA
26 SABRE(TOOTH) - thanks to Duncan for the reminder - I found this but didn’t record it properly.
35 (CUSPID)ORE
36 (CANINE) DISTEMPER
4 NEW(FANG)LED
8 ADVENTURE(GAM)E
13 LEXI(COG)RAPHER
21 (NIPPER)KINS
29 (IVORY)TOWER

Three treatment items: ELEVATOR, FORCEPS, GAUZE - all of which can be used when extracting a tooth.

I haven’t analysed many clues below - ask if I’ve missed one that puzzles you

Is is just me being dense, or was this rather difficult today? I could do with some help with explaining 18d (HACKSAW). That said, there are lots of marvellous surface readings here (too many to pick out any particular favourites) that made it very enjoyable, albeit time-consuming.

I think part of the reason I had trouble was from some slightly unusual devices being used: e.g. some rarely used abbreviations (”the same” for DO, “leaves” for FF, “rod” for R, etc.) and apostrophe-S being used in some of the subsidiary parts.

I found this much tougher than most Phi puzzles and two clues (18 and 22) had me completely stumped as the wordplay in each of them led to an answer that didn’t really seem connected to the definition. Thanks to nmsindy for explaining both of these to me.

I enjoyed this one a great deal. I had the grid filled in reasonably quickly but then had to go over half a dozen or so clues where I’d filled in the answer without fully understanding what was happening. A mistake at 16 held me up but not for too long and 11 was the last to go in and also the last clue I deciphered.