Is it permissible to wipe over a cast and wash whatever can be washed, leaving out what is near the cast?

What you did was incorrect. You were obligated to wash all exposed skin and not leave any exposed part unwashed. You could have done this while exercising caution to avoid harming the cast. This prayer of yours is invalid, and you must repeat it according to the majority of scholars. However, some believe that there is no need to make up the prayer if you were unaware of the ruling. The opinion of the majority is more cautious and absolves one's responsibility.