Q.What to do next with my dying pussy willow tree / new willow tree.
Do I cut the new stem away and plant it or should I cut away the dead tree leaving the new stem on the base of the trunk thanks

Your tree is grafted at the top to create this standard tree. The branch growing from below the graft line will not be a Weeping Pussy Willow, most likely.
I would check with the garden center that you purchased the tree from; most reputable growers will guarantee a new tree for 1 year.
https://www.gardeningknowhow.com/ornamental/shrubs/pussy-willow/weeping-pussy-willow-care.htm