Do you need the cover at the moment? I would ditch the cover for a while and see how the pool does.
Are you brushing weekly? Pools need a good brushing weekly.
I agree with ping! And you stated:
Is is simply a case of my filter is not big enough or I'm not running it long enough even with the cover on?
You weren't running the filter as much with the cover on which can create *dead spots* in the pool. I'm not surprised it went green. It is all fixable though.