If Ghusl is fard, after intercourse, do we have to remove dry mucus from the nose as well or would just pulling water up to the soft bone be sufficient
One of the conditions for the validity of wudhu and ghusl is that water must reach the area which is required to be washed for the wudhu and ghusl to be valid and correct. (See Nurul Idah p.31)
Regarding your question as long as the water reaches the nose the ghusl will be valid. There is no need for you to remove the mucus before starting the ghusl.
Only Allah Knows Best
Mohammed Tosir Miah
Darul Ifta Birmingham.