What is the ruling regarding someone who performed ablution with the intention of the obligatory washing of the hands, but did not wash the entire hand, and then found it difficult to make up for missed prayers, so he started making them up with the worshippers in the mosque?

If you perform ghusl (ritual bath) with the intention of wudu (ablution) and suffice with washing the hands without the forearms, your wudu is valid according to the Maliki and Hanafi schools, and consequently, your prayers are valid and you are not required to repeat them. However, if the wudu was invalid, you must make up the missed prayers. The manner of making them up is to pray daily as many of the missed prayers as you can until you have compensated for all that is due, provided that this does not result in physical harm or disrupt your livelihood. The making up of prayers must be done immediately unless the person is harmed. Regarding making up a missed prayer behind someone who is performing a different current prayer, there is no harm in this according to some scholars.