[Argus25]

No,

North must be associated with north and south with south, you are trying to reinforce the existing field from a location outside the magnetic gap.

You wind the coil around the circumference of the assembly, not spilling over the edges of it, maybe 1/2 to an inch wide, but no wider than the focus magnet assembly or the plates that secure them.

You just need to be 100% sure that the applied voltage polarity and wire turns direction results in a magnetic field from the coil that aids the polarity of the N & S poles generated by the focus magnet, not opposing it or sticking to it, but repelling it, or you will end up trying to reverse magnetize them and grossly weaken them.

It seems counter intuitive because N & N and S & S poles repel each other. But it makes more sense if you think of the polarity of a focus coil which achieves the same effect as the magnet and will magnetize a magnet, with that polarity, think of the magnet as having a memory for the applied field from the coil.

So unless you are confident that you can 1) identify the poles correctly on your focus magnet, and 2) calculate the expected field from the coil, it is better to set up a dummy coil with some turns direction and a known applied polarity from the battery (with a current limiting resistor) and check with the compass that the poles of the coil and magnet assembly match before you either wind the wire around, or slip the coil over the magnetic assembly and apply the high current.

I wound the wire tightly over the assembly, with the high magnetic repulsion forces when the current is high, the coil could jump around/off the assembly if not properly secured.

(Another way to weaken or calibrate the field downwards could be with the same coil, with stepwise lower pulses of current in reverse, rather than having to weaken the magnets with mechanical disassembly)

PS: Of note there are other devices you might work on where disassembly of components weakens magnets. One is the DC brush motor, when the armature is removed, but if you are wise to it you can slide in a keeper. (A classic for this is the Capstan motor in the Nagra MK2 professional tape recorder) Likely I think, these devices, in many cases , like the PM focus magnet assembly for CRT's, are magnetized after they are assembled at the factory, which explains why the fields are weakened after dis-assembly and reassembly.