Dying bleach damaged hair advise

If the dye is an oxidative one, ie it needs to be mixed with peroxide (developer) it will further damage the bleached areas and there is a chance it will break off. Bleach damage is cumulative, so any further processing could be the straw that breaks the camels back.
Even if your hair is strong enough, permanent dyes will wash out very quickly on bleached hair.
I would avoid any more chemical processing and use only direct dyes which are totally safe and non-damaging. If you dye the bleached parts darker your roots won't be as obvious. I would use something like Adore, which comes in a range of natural colours. Medium Brown and Mocha can be mixed and diluted with conditioner to make various shades of brown and blonde. (The one thing they don't make is pre-mixed blonde direct dyes, I really wish they would)
Wella colour Fresh is a range of direct dyes that does include blonde shades, but they won't last a s long as Adore.

Thank you I shall give it a go