I've read through the information I could find about Euonymus problems but I'm still not sure what is wrong with mine! :?
It has narrow white hatch-type marks all over it (almost like larvae but they do not move), has lost a lot of leaves and twigs and appears generally unhealthy.
Does anyone know what this problem is and how to treat it?
Sounds like Euonymus scale aka Euonymus alatus scale.* You can read about it here.
There is a clickable link to this page, but I would never use most of the toxic chemicals they recommend.*
They do recommend insecticidal soap and horticultural oils, both of which would be a good option.* If you use a horticultural oil I would recommend one such as Organocide, which is made from sesame seed oil and NOT Organicide which is made from petroleum.