Fitting An Axe With A Handle
If the head is loose you can soak the axe for a few hours in a bucket of water but this will only ever be a temporary fix (the wood will swell and the handle will be tight for a little while). Ideally you need to replace the handle or at least replace the wedge.
Saw off the old handle close to the axe head (Note: you should not just burn the handle off as this will cause the bit to lose its temper, note the above arrangements) and drill out the wood that is left in the axe head. You can punch the rest of the wood backward from the handle side and then clean inside the hole.
Get the new handle and pound it into the head. Make sure the handle passes out the other side by at least an inch. Also be sure that the new handle is correctly aligned.
Put some glue on a wedge and in the slit for the wedge in the end of the handle (you did make a slit didn't you?) and then drive in the wedge. You can now saw off the excess handle and wedge.
It would be a good idea to fix the wooden wedge in place with a steel wedge, placed diagonal to the wooden wedge.