You are right about the second example:
... mit dem sie ihre Abende und Wochenenden verbringen können.
is indeed correct.
The first example:
... mit dem sie ihre Abend und Wochenende verbringen können.
is not correct for two reasons:
First, it is "der Abend" and the possessive pronoun ("ihr") has to reflect that: "... sie ihren Abend ...".
Second, and this is probably hard to understand for a non-native speaker: the difference between the two examples is that in one the singular is used, in the other the plural. In singular the "Wochenende" needs an article. Here is the corrected phrase:
... mit dem sie ihren Abend und das Wochenende verbringen können.
I cannot quote any rule or source for that, but without the article it just "sounds wrong". I suppose that the possessive pronoun already implies the article (a case of the so-called "Nullartikel") and this is the reason why it can foregone, whereas "Wochenende" still needs the article because it is not covered by the "ihr".
If you wouldn't use the possessive pronoun both nouns would need an article:
... mit dem sie den Abend und das Wochenende verbringen können.