According to both the dictionary (Pleco) and the grammar rules (Chinese Grammar Wiki) it seems that 有 (or 没有) should typically be used in a combination 有 + object and expresses either possession or existence. Yet, 有 appears to be also used before a verb. Here are 2 examples:
In the song "好想你" by Joyce Chu there is a line "但却总爱问我有没有想你" which is translated as "Yet you love to ask if I missed you" (see e.g. lyricstranslate). In this case the English translation does not have any words indicating possession. Naively (at my beginner Mandarin level) I would translate the English phrase back to Mandarin as "但却你爱问我想不想你" (i.e. without 有).
In a begginer-level Mandarin story about a cat (一只猫) there is a line "前天天气不好，我没有吃东西；" I can guess the meaning (The day before yesterday the weather was bad and I (cat) didn't have anything to eat), yet I do not understand how it is constructed grammatically because 吃 (and not 东西) follows 有.
What is the grammatical structure and the function of 有 in these sentences?
For completeness, here is my (possibly incorrect) attempt to understand what is going on here. One conjecture for the example #2 is that the structure is actually verb (没有) + object (东西) and 吃 is used as an adjective for 东西. If I try to apply the same idea to #1 I would have to conclude that 你 is an object and the literal translation of "我有没有想你" would then be "Do I possess you to miss?".