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Despite re-reading all the comments I still can't get 13a, 24d or 27d despite having all the crossers so more nudges would be appreciated.

Hi Tintin
13a is a classical allusion. Try the standard abbreviation for the first word followed by the name of a sign with "one" removed.
24d abbreviation for the first two words followed by "longing" upwards
27d first letter is indicated by first three words, then a short wotd for "appeal", and finally remove a short synonym for "when" from the last word.
Good luck!

Many thanks for your generous hints BuzzyBee - all done now! I had already tried what you suggested for 13a but the resulting word meant nothing to me (I've never been good on mythology!)

I have everything except 22d, for which my crossing letters are J?E?E?. That suggests an answer but I can’t reverse engineer a pair from the clue.

Jack, the usual 'stone' goes inside the 'one trying to win... someone's affections'.

D’oh! Thanks Malone!

Glad I could help, Jack.

Finished at last with 24d the final one to be entered, but I am puzzled by Buzzybee's hint to Tintin containing "longing". The clue in the version I printed from the website reads "Head office ready to be turned over". Is there a version which has "head office longing to be turned over"?

I'm pretty sure there's only one version of the clue at 24dn. Tintin's hint will get you to the clued answer. I took "ready" to be monetary.

My version is (was) definitely "Head office longing to rise". I live in New Zealand (check out the origin of my nom de plume), and am awake and ready to go when the puzzle comes out at midnight UK time. I therefore tend to see the published version before any changes/corrections.
BuzzyBee