When the body produces antibodies, can these antibodies deal with small mutations of their corresponding pathogen?

Sometimes they will, sometimes they won't - it all depends what changes, if any, a mutation may make.

In some cases they do. Many years ago there was research done on Phages. However, these are so small that's it's difficult to find the right one. The theory was when a bad virus entered the body and the proper phage was introduced; the phage would eat the bad virus. The bad virus would evolve and so would the phage in a war of escalation. Fascinating stuff. However, finding the right phage is about the same as finding the virus....real tough to do.

It depends on the amount of mutation.  There could still be enough proteins on the surface of the virus for the antibodies to recognize and attach to.  If the mutation is severe enough, then it's just like a new virus.  this is what has happened to Covid-19, it's mutated enough that our bodies haven't seen it or something sufficiently close to it to mount an effective defense.  Now, our bodies do learn how to fight it but sometime our immune system goes overboard and starts to attack healthy cells, that's what is causing the problems here.

They have to do a different flu vaccine every year to address the types of flu they predict will be prevalent.  So I would imagine that mutations would probably require a change in the vaccine to be more effective.