As a learner I think you can use that and when. However, there are two points:
Regarding the terminology, that is a relative pronoun, but I am not one hundred percent sure that if we can call where and when a relative pronoun as Cambridge claims. Generally, I think since when and where cannot act as the subject, it's not a good idea to call them relative pronoun.
I learned to call when, where, after, and so force subordinate conjunction which are usually used to construct an adverbial clause.
In defining relative clause you can use that instead of the relative pronouns whom, who, and which while you cannot do so in non-defining relative clauses.
So, I think the sentence can be considered as:
If there’s a time which(that) the grammys favor white people, I hope it’s this time.
The other point is that replacing the relative pronoun with that makes the sentence less formal and it's more likely in conversations.