Phoenix

[fuzzy-brain]

Last two.
24a. Before tea, sounds like idiot joins late Houston singer to make flight possible. (5) P _ A _ T.
29a. Being confused, in other words, lightly burns body hair of rock band. (6) G _ E _ S _ . I answered 'Gneiss' (a rock), and Genesis is a rock band - took out an 'e' but cannot parse.

[JJ109]

29a Lightly burns = singes (anagram of gneiss), but......

[Lie-in King]

24a Prat-t (& Whitney [Houston] - make aircraft engines)

JJ has it - 'singes' (in other words, "lightly burns body hair") is confused (anag) = Gneiss (rock band)

This is an oddly-clued crossword...

[fuzzy-brain]

Thanks, jj109 and LIK.
LIK, if 'singes' is correct, then your earlier answer 'Ships Log' (the Mercury clue) doesn't fit as 29a would then begin with 'G'.
I'm confused!

[Lie-in King]

Apologies, I worded my answer badly - Gneiss is correct

[fuzzy-brain]

Thanks once again, LIK.