A great but somewhat tricky puzzle this morning.
A number of clues seemed fairly straightforward but took us a bit of time to then be able to describe exactly. Nice to see a nod to my (half of Teacow) hometown! (PRETORIA). Some really good clues – we particularly liked FATHOMED, HAIRCUT and VIETNAM.
Not 100% sure about OPERATION for influence but the parsing fits perfectly.
Thanks Armonie!

Across
1 Judge a prince’s eulogy (8)
APPRAISE
A + P (prince) + PRAISE (eulogy)
5 Bob has to fasten accessory (3,3)
BOW TIE
BOW (bob) + TIE (fasten)
9 Intellectual finds bromine in food (8)
CEREBRAL
BR (bromine) in CEREAL (food)
10 Reproach worker trapping vermin (6)
BERATE
BEE (worker) trapping RAT (vermin)
12 Holy woman to beg for shower (5)
SPRAY
S (holy woman) + PRAY (to beg)
13 A graduate’s coming in to rework stone (9)
ALABASTER
(A + BAS (graduate’s)) coming in to ALTER (rework)
14 Checks around a hundred Fiats (6)
EDICTS
EDITS (checks) around C (hundred)
16 Husband’s manner stung? That’s distressing? (7)
HAIRCUT
H (husband) + AIR (manner) + CUT (stung)
Distressing in the sense of removing tresses
19 Enthusiasm when circuiting volcano in the country (7)
VIETNAM
VIM (enthusiasm) circuiting ETNA (volcano)
21 Small car’s no good in heavy industry (6)
MINING
MINI (small car) + NG (no good)
23 Cute if led astray becoming two-faced (9)
DECEITFUL
(CUTE IF LED)* (*astray)
25 Rare beast for the Godfather? (5)
PANDA
PAN (god) + DA (father)
26 Name complete idiot (6)
NUTTER
N (name) + UTTER (complete)
27 I report a disturbance in South Africa (8)
PRETORIA
(I REPORT A)* (*disturbance)
28 Win when giving complete attention (6)
ENDEAR
END (complete) + EAR (attention)
29 Understood fellow to be comfortable with daughter (8)
FATHOMED
F (fellow) + AT HOME (comfortable) + D (daughter)
Down
1 A County Council to take advantage and press charges (6)
ACCUSE
A + CC (county council) + USE (take advantage)
2 Bird to fly from rooftop (9)
PARTRIDGE
PART (fly from) + RIDGE (rooftop)
3 French priest unknown in monastery (5)
ABBEY
ABBE (french priest, Abbe Pierre) + Y (unknown)
4 Soprano gets each singer fish (3,4)
SEA BASS
S (soprano) + EA (each) + BASS (singer)
6 Influence speech about training (9)
OPERATION
ORATION (speech) about PE (training)
7 Child eats a bit of sausage in grill (5)
TOAST
TOT (child) eats (A + S (bit of sausage))
8 Condemn old banger to bottom of river (8)
EXECRATE
CRATE (old banger) to bottom of EXE (river)
11 Baldric’s son Rowan (4)
SASH
S (son) + ASH (rowan, tree)
15 In Spanish bar the French demonstrate vocal style (9)
CANTILENA
in CANTINA (spanish bar) LE (the french)
17 Care of sister’s instrument becoming a puzzle (9)
CONUNDRUM
CO (care of) + NUN DRUM (sister’s instrument)
18 Suspect vice? Need proof! (8)
EVIDENCE
(VICE NEED)* (*suspect)
20 Mishandle a hand warmer (4)
MUFF
Double Definition
21 Girl touring US city gets fever (7)
MALARIA
MARIA (girl) touring LA (US city)
22 Thanks poet for jacket (6)
TABARD
TA (thanks) + BARD (poet)
24 Quoted detectives arresting Lawrence (5)
CITED
CID (detectives) arresting TE (Lawrence, T E Lawrence of Arabia)
25 Stroke companion in bed (5)
PATCH
PAT (stroke) + CH (companion)
Nice. I particularly liked FATHOMED. Like you, I was surprised that OPERATION could mean ‘influence’, but this is definition 4 in my Chambers app.
Thanks to Armonie and Teacow.
Thanks Armonie and Teacow. A good puzzle and most went in smoothly, but ENDEAR took me some considerable time to get.
While I was pleasantly surprised by the lovely charade of FATHOMED, this clue was actually the only one of which I thought “hmm, that’s not right”.
I don’t think there is any justification for Armonie to use ‘to be’ here.
Other than that, a nice work-out in his familiar style.
Many thanks to Teacow & Armonie.
Thanks Armonie and Teacow
At the harder end of this setter’s scale but all very gettable with a number of clues holding me up for longer than usual – the S (what I expect is the abbreviation of a ‘saints’), the logic behind the wordplay of PANDA and had to look up ‘baldric’ at 11d.
Some nice touches and very rarely is it that you won’t find one of his unusual synonyms verified in the dictionary. I particularly liked his construction of VIETNAM, NUTTER and CONUNDRUM.
Finished with that SPRAY and SASH as the last two.
Thanks to Armonie and Teacow. Enjoyable. I took a while parsing PANDA and was surprised by ridge = rooftop in PARTRIDGE.
I wouldn’t describe this as tricky, I’d describe it as bad. Very bad. 6,8,&11 are unforgivably bad. So is 16ac.