Synonyms for Horn in:

n.

enter, cut in. participate, interlope, meddle, butt in, interfere. horn in (noun)
meddle with, intrude into, poke into.

v.

get involved (verb)
butt in, meddle with.
interrupt (verb)
butt in.

Other synonyms:

interlope, enter, meddle. interfere. interrupt
cut in.
Other relevant words:
meddle, interlope, interfere, enter.