To be honest with you if it looks crap on the surface there's a good chance it will crap all the way through. Grinding the surface flush and welding over the top isn't gonna change owt, remember, you can't polish a turd
If the surface finish of a weld looks poor, its generally a good indicator that the weld is rubbish too. Welding over it again is doing nothing to help it, other than producing a finished weld that looks like a great big dead slug.
Far better to grind the whole lot out and start again; a horrible job but makes you think a bit harder about producing a decent weld in the first place.
There's nothing wrong with giving a weld a quick tidy with a flap disc or grinding disc to tidy up corners/stop-starts etc, but not a full-on scale and polish
I can remember my Guru (well almost) Big Bill James.
An engineer I have been asking for advice for years asked me 'Will you be grinding your welds out' and I said 'Not if they are ok' he nodded and stomped off.
I think it was a compliment, and a useful hint. If a weld is any good it does not need to be concealed.
As a sculptor I think I am making welded sculpture, why should I hide my welds.
I confess if they are crap I might grind the odd one out a re-lay it. but a weld is a weld and should be something to be proud of. I also hate grinding.
A job done well should look good even if people cant see it.
Has that helped? rereading the original post probably not.
If it is structural and looks a mess as welderpaul says dig it out and do it again. A good way is to use 3mm cutting discs, you can get right back into the sucker.
But dont grind them off for cosmetics, nice welds look fine and are totally honest.