That's a colour purity problem where the CRT's Red, Green and Blue electron beams are being distorted by a magnetic force and not hitting the correct phosphor dots for that colour.

If you really can't eliminate the source of the magnetism causing this, it might be possible to have the purity ring magnets on the neck of the CRT adjusted to correct this, but it's probably got to be your last resort at fixing this. It's perfectly OK to mark their positions with a felt tip pen and adjust them (and only them) to see if you can fix this yourself - purity faults show best on a red background BTW).

I'm sure that I don't need to warn you about the 20kV feeding the CRT - don't be earthed while doing any of this and keep your left hand in your pocket while working with your right.

Adjusting these magnets may cause convergence errors that will also need to be corrected - convergence is where all 3 electron guns are calibrated to hit the same 'spot' on the CRT at once to produce white (white being a mixture of RGB all at once, for example). If the convergence is out then you will see red, green or blue borders for white objects where some of the colour guns are misaligned. If you get this far and are bothered by convergence errors then call a TV engineer and see if they're prepared to come and adjust it.

As has been pointed out already in this thread, it is possible to "ruin" a CRT with residual magnetism. The "ruined" CRT will have a magnetised shadow mask (the 'gauze' behind the glass front) or a magnetised rim band (the steel mounting/reinforcing band around the extremities) that the degauss coil is unable to demagnetise. A TV engineer can use a handheld degaussing wand to manually demagnetise the screen and that device is an order of magnitude more powerful than the inbuilt degauss circuit but there's still situations where it isn't possible to degauss sufficiently without removing the CRT from the cabinet.