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29a. Likely to drop off fencing equipment to the front yard. (6)
DROWSY please help with parsing.

A reversal of SWORD (fencing equipment) + Y (yard)

Sword, fencing equipment, reversed, to the front - Drows+ add Y, yard.

Duh! Don’t spot the reversal, thank you.

39d. Clipper shaved Katy's last bit of hair off (5,4). I have C?T?Y ?A?K so it must be Cutty Sark but I can't parse the rest apart from tys at the end of Katy's.

Cut = shaved
tysark - anagram ('off') of Katys and the R at the end of 'hair' (last bit of hair).

Thank you for the prompt and useful reply!

Dicki, you're welcome.

19a Drifters work out what to do with four letters from the 11th (8)
?L?N?T?N
Plankton, as a rather arch definition of drifters?

I can almost but not quite parse this one:
35a Tanker with late British figurehead on front of ship (6)
Bowser, I think. Er = late British figurehead. Front of ship is either bow, then how is the 's' explained, or it is the 's' (front of ship), then how is the bow explained?

SB, for 35 Bows is the 'front of ship' - the Chambers definition expressly says that the plural is often used.

19 Yes, I think it's just a rather arch, whimsical definition.