I agree with Wayne. The "HAZ" (heat affected zone) on welded, hardened steel is unavoidable. Pre-heating just helps to avoid quenchback cracking.
That's why welding on an anvil is absolutely a last resort and generally to be avoided unless the anvil face is separated or so cratered as to be completely unusable.
I had major torch cuts in mine and had little choice, but even then i prepped fastidiously, pre and post heated, etc, and did as little welding as possible.... Now, many years later, i'm beginning to see some hairline cracks in the HAZ.
The deed being done, i use a very hard, polished, rounding hammer ( slightly domed face) to gently work dings back to flat...The same technique works on the horn too. The advantage is that the soft area gradually work hardens. The drawback is that it'll eventually get too brittle and crack.