Is it permissible to pray with a "darnaga" in the mosque?

There is no harm for your father in praying with the splint as long as he is harmed by its removal. It is preferable for him to perform ablution for each prayer after its time begins, to avoid the disagreement of those who consider ablution invalidated by the exit of a large amount of blood. There is also no harm for him in praying with it in the mosque, provided that he ensures it will not contaminate the mosque. This is supported by the permissibility for a woman experiencing irregular bleeding (mustahadah) to enter the mosque and pray in it if she ensures no contamination. The Prophet, peace and blessings be upon him, once performed i'tikaf (spiritual retreat in the mosque) with some of his wives, one of whom was experiencing irregular bleeding, and she would place a basin beneath her to collect the blood. Al-Shawkani stated that this hadith indicates the permissibility for a mustahadah woman to stay in the mosque, and the validity of her i'tikaf and prayer. This ruling extends to those who experience continuous discharges and those with a bleeding wound.