Is it permissible to perform all prayers while having a patch over the male's urethral opening to prevent urine leakage after urination, and suffice with performing ablution for each obligatory prayer without performing istinja' (washing the private parts with water), in order to avoid the hardship of performing istinja' for every prayer?

If the discharge of drops has a time when it ceases, then one must wait for it to stop and then perform istinja (definition) (cleansing). Placing an adhesive patch on the urethral opening immediately after urination is not sufficient if you know that urine is still being discharged, because if the discharge flows and the patch becomes wet, the ablution (wudu) is invalidated, and istinja becomes obligatory. If you perform ablution before performing istinja, the ablution is not valid. However, if you are certain that nothing is discharged, then your ablution is valid.