Handy part under which there is domination (5)

I believe the answer is:

thumb

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'handy part' is the definition.
'thumb' can be an answer for 'part' (thumb is a kind of part). I am unsure of the 'handy' bit.

I can't explain the remainder of the clue.

Can you help me to learn more?

(Other definitions for thumb that I've seen before include "Travel by . . . . . - hitchhike" , "Try to get a lift" , "Hitch ride" , "Attempt to obtain a lift" , "One opposed to other manual workers" .)

I've seen this clue in The Irish News.

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