Yes, there are much better things on to use, even off the WalMart shelf.
Bleach not only burns like the dickens but it is caustic and will damage healthy tissue.
Unless there's some serious fungal/bacterial stuff going on in the frog, Absorbine Hooflex Thrush Remedy is an excellent treatment, serves well as preventative maintenance, does not damage healthy tissue, and can be rubbed on the heel bulbs/fetlock joint to help prevent scratches.
Tractor Supply carries it and many local feed and tack shops also carry it.
Hooflex® Thrush Remedy – Hoof Care – Products – Absorbine
A little goes a long way and lasts for several days, even in mud
As far as minor cuts, just grab the Polysporin, Neosporin, Triple Antibiotic, or whatever's in your bathroom cabinet, and apply to the cut after gently cleaning.