Though Henry Mulholland had a son and a grandson named Beresford Mulholland, I have found no evidence that his grandfather or his father Richard married a Beresford.
The other possible connection with the Beresford name would be from Henry's wife, Maria York, but as of yet, I have not found her connection with the Beresfords.
I have an Alexander cousin who married a Beresford.
David V..Hughey, Ph.D., descended from Capt. John Alexander, Jr. of Enagh and Caw