Translation:During summer, she attends the mosque with him most of the time.
"In summer, she usually goes with him to the mosque." When possible, I find "usually" works in place of "most of the time" or "in the majority of cases". From an English perspective, simplicity provides a clearer understanding. But, I'm not a language specialist, just a language user.
Thats what I put too, because that is what many/most English speakers would say, or perhaps "...she usually goes to the mosque with him". "During summer" sounds a bit strange to me. "in summer" or perhaps even "over summer" would be better here. "Attends" is not wrong but would not be the most common word used in this situation, particularly as the German verb gehen is used. Also "usually" is a perfectly good translation of "meistens" according to dictcc
In summary, your answer is far more likely to be said by an English speaker and should not be marked wrong in my opinion. Reported.
As a native German speaker: The given translation for "During the summer, she attends the mosque with him most of the time" is not wrong but I want to give a translation which is closer to the English sentence ... in my opinion: "Während des Sommers besucht sie die Moschee meistens mit ihm".
To me these two sentences have a slightly different meaning. When reading the German sentence, I thought of "her" who does all sorts of things with "him". Maybe skiing in winter and travelling in spring. And in summer, she usually goes to the mosque with him.
Now the English sentence makes me think of "him", who goes to mosque all year round and of "her" who accompanies him only in summer.
Probably you can read both meanings into both of the sentences, still leaves me a bit puzzled.