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You should have TA in 13a. Short word for deTECtive with an I inserted.
I get 2 spare Ps in 20. tHIs and OS.
This seems to work. Just noticed a nice clue to the theme in the 22 answers.

I also have an A in 13ac (left) but still don’t understand how to join up.
The order suggested matches the letters I have but no pdm is coming!

Google the two answers to 22 (together) as they appear in the grid.

Doesn't 20across have an extra letter in the wordplay on the right side?

Yes it does om1957, it contributes to the instruction I think.

There are two unclued Ps in 20 on the left so from the preamble the right entry must be normal. I make it an anagram of abrasion without BOTH ends.

That makes sense, the a needed for the instruction must come from somewhere else?

The A of the instruction comes from an extra letter in a clue in 12ac

Thanks murky, I had omitted letters in that one (incorrectly). My 'curved line' works much better now!

Thanks smithsax and murky. My puzzlement now explained.