He threatens the kid to finish the work.
Who is performing the action "finish the work",
He or the kid.
He threatens the kid to finish the work.
Who is performing the action "finish the work",
He or the kid.
He threatens the kid [to finish the work.]
This sentence is ungrammatical because the catenative verb "threaten" can't take an infinitival clause as complement when there is an intervening noun phrase (here, "the kid") between the two verbs.
"Threaten" can occur freely, though, with to-infinitival complements in simple catenative constructions, i.e. the kind without an intervening noun phrase:
He threatened [to leave].
She threatened [to tell her father].