Is there a way to move a black hole?
In addition to what @Jack Mazy said about force due to gravitational attraction, there are also charged (Reissner–Nordström) black holes. I would imagine in theory you could take another charged object and place it near the black hole which would cause Coulomb force on black hole putting it in motion.
Yes, it is perfectly possible to move a black hole, in any reasonable sense.
In particular something that is important to understand is that, far from the event horizon, black holes are not particularly special. The gravitational field of any spherically-symmetric distribution of mass is the Schwarzschild solution in the region where there is no matter for instance: the gravitational field of the Earth, say, is rather well-approximated by it.
Another thing that is important to realise is that, when fields are weak and relative velocities low, GR is very well-approximated by Newtonian gravitation. So, far from a black hole's horizon, they really don't look very weird, gravitationally.
So if you can move a planet, or a star, you can move a black hole: you just bring some other mass near it and gravity does the rest. Alternatively (and, really, more easily) you can crash some initially-slowly-moving other object into it and (although the details of the collision will be hard to understand as it will be highly relativistic) you can stand a long way away and just do boring old conservation of linear momentum, and discover that the velocity of the black hole has changed.
Note I have intentionally skated over some details here.
As a person before me commented, white holes might not even exist, so I won't talk about those. A black hole is a super condensed ball of mater and light, so gravitation physics remain the same. If you were to put a more massive black hole next to it,they would both converge on each other, causing then to move through spacetime with all of its contents. If you really want an answer on white holes though, they would repel each other and not attract.