This is Lady Susan Beresford and her horse (with no name). Susan was a daughter of Lady Blanche Somerset (daughter of the 8th Duke of Beaufort) and John Henry de la Poer Beresford (fifth Marquis of Waterford, Lord Lieutenant of Waterford and Honorary Colonel of the Royal Artillery, South Irish Division).
At this time the Beresfords lived at Mayfield House, Portlaw, Co. Waterford. Here's some information on Mayfield House, unfortunately now in ruins...
Date: Saturday, 15 December 1900
NLI Ref.: P_WP_1131