Can the sentence
I hate Jill singing those songs.
I hate Jill when she is singing those songs.
Or does it mean something else?
"I hate Jill singing those songs" could only be interpreted as meaning something like "I hate it when Jill sings those songs", "I hate that Jill sings those songs", or ""I hate Jill's singing those songs". I agree with Chris H's answer that "I hate Jill singing those songs" could not practically be interpreted as meaning "I hate Jill when she is singing those songs."
I'm not sure yet of the exact explanation for why this interpretation is ruled out. It does seem to be possible to interpret the similarly constructed sentence "I hate the girl singing those songs" as meaning "I hate the girl who is singing those songs".
I think it may be related to the phenomenon called "whiz-deletion". In some cases, postnominal participles are analyzed as having some kind of relationship to wh-relative clauses using a form of the auxiliary be.
When we look at similar sentences with explicit integrated relative clauses of this form, "I hate the girl who is singing those songs" is fine, but "I hate Jill who is singing those songs" doesn't seem right.
Something seems to be going on with information structure here: in a sentence like "I hate the girl who is singing those songs", the clause "who is singing these songs" is what would traditionally be called a "restrictive" relative clause, in that it communicates some information about "which girl" the speaker is talking about, whereas in a sentence with a proper noun not preceded by a definite article like "I hate Jill...", the proper noun by itself completely specifies which person we are talking about, which seems to preclude the use of an integrated relative clause or of related postpositive constructions that can be analyzed as being derived from relative clauses by "whiz-deletion".
For example, I have a similar feeling about the sentence "I hate the girl in the green hat" (it seems OK, and means "I hate the girl who is in the green hat") vs. "I hate Jill in the green hat" (it sounds odd, and feels like it might mean something like "I hate it when Jill is in the green hat" or "I hate having Jill in the green hat").
In typical use, you can rule out this interpretation. Reading Jill singing those songs as a noun phrase is much more natural than reading it as Jill (when she is) singing those songs.
On reading or hearing that sentence your audience would understand your hatred to be of the action not the individual. If you wanted to give the other meaning you would have to make explicit the when she sings those songs (or ...is singing...)
The two following sentences are equivalent:
The subject of both sentences is I. The verb of both sentences is the transitive verb hate. Both sentences use a non-finite verb clause as the direct object of hate, and both of those non-finite verbs have the same subject of Jill.
In the first, that non-finite verb phrase is a to-infinitive clause. In the second, that non-finite verb phrase is an -ing clause. Both those two non-finite verbs have Jill as their subject.
That means that Jill is not the object of hate: the entire non-finite verb clause is. Jill cannot be the object of hate given that she is the subject of the other verb, respectively to sing and singing.
The only difference between the two is that when you provide the non-finite verb clauses with their own subject, the to-infinitive needs a for-complementizer, whereas the -ing clause can take a bare subject without the complementizer.
In short, this says that it’s the singing you hate, not Jill herself. Your proposed reading that you hate Jill only during those times when she is singing those songs doesn’t follow from the grammar.
Both constructions are to be avoided, because Jill becomes the object of your hate, whether it is "Jill singing" or "Jill when she is singing." It isn't Jill that you hate, so let's not make her the object of hate. It is more common to say "I hate it when Jill sings those songs." The undefined "it" becomes the object of hate, and your love for Jill is undiminished, but you're annoyed or aggravated when she does a particular thing.