manure on the top of the ground

I covered the soil on raised beds on 20th December 2017 I have now planted potatoes in these beds about 6" deep.   Do you think that the shaws will be burnt by the manure as they push through the soil?    The manure has been laid out in a Farmer's field for 3/4 months before it was transported and laid on my soil.

No, they'll be fine. The rain will have washed through the manure to neutralise any corrosive contents. I threw fresh horse manure over my Asparagus bed in Feb and the asparagus has come through just fine. I think you may have had even more rain than us.