Is it permissible to perform Hajj on behalf of a deceased brother this year, given the presence of transparent, viscous secretions that emerge after urination and necessitate washing clothes for each prayer, or is it better to postpone Hajj until treatment? And are these secretions considered madhiy (pre-seminal fluid), wadiy (post-urination discharge), or something else, and what is their ruling in each case?

What often comes out after urination is wady, which is impure and invalidates ablution (wudu). It does not necessitate a full ritual bath (ghusl); rather, the male organ should be washed thereafter, and ablution performed. Since these drops only descend after urination, you must prepare for prayer sufficiently before the prayer time begins, allowing for their cessation. Use a piece of cloth to prevent the impurity from spreading to your clothes. If you forget this or find it difficult to change your clothes, there is no harm in praying in them. It is not permissible to pray while the drops are descending if you know they will cease, even if this leads to missing congregational prayer. As for performing Hajj on behalf of your brother, there is no harm in delaying it until next year if that is easier for you.