Single word for "embrace and extend and possibly corrupt ideas of a political movement"

Perhaps you want the word co-opt.

Merriam-Webster gives:

a : to take into a group (as a faction, movement, or culture) : absorb, assimilate. the students are co–opted by a system they serve even in their struggle against it — A. C. Danto

b : take over, appropriate. a style co–opted by advertisers

Appropriate (as a verb, pronounced with a long final a, to rhyme with state) may also serve your purpose.

Recuperate is almost always used to mean "to recover from an illness". Its alternate meaning of "to recover something that was lost" doesn't fit here either. Merriam-Webster also gives "to bring back into use or currency : revive" but that seems very uncommon, and is no better.

If you want to focus on the "corrupt" portion, you could use bastardize:

to modify especially by introducing discordant or disparate elements
_{from m-w.com}

Also to to adopt may fit you description:

• To take up and make one's own: adopt a new idea.