something i've wondering, does a jet like this "push" the blackhole in any way?
That being said, momentum is conserved, so if there was only a single jet then the "black hole system" minus the jet would have momentum. However, I don't think we've ever seen a single jet black hole.
Also keep in mind we have a rather poor understanding of black holes, and that a lot of information that is stated as "we know X about black holes" is actually "general relativity equations evaluate to X" and has not been observed or verified by data. There's still serious open questions about their physics.