Just like you observed, some native speakers, especially the ones from the south, say 我有看 instead of 我看了 . They also often say 我有看过 to mean "I have the experience watching (it.)"
However, using 有 before a verb is not considered standard, as it is a dialect influenced structure. If you teacher wish to stick to the standard grammar, both 我有看 and 我有看过 would be considered "bad grammar."
That being said, as a native speaker and a Mandarin teacher from Taiwan, I use these structure every single day with my friends but refrain from using them with my students who want to learn standard grammar.
Here is a sample dialog with standard grammar:
A:"你看书了吗?“ (Did you read the book?)
B: “我看了。” (I did.)
Here is a sample dialog of roughly the same meaning using 有 (not standard!!!!)
A:“你有看书吗?” (Did you read the book?)
B: “我有看。” (I did.)
In this particular case, the 有 used by some speakers is a rough equivalence to verb + 了, signifying completion and/or occurrence of an action.
You mentioned 过 in your question. 过 is used when talking about past experiences, not completion. Here is a sample dialog:
A:"你看过这本书吗?“ (Have you ever read this book?)
B: “我看过。” (I have.)
Here is a sample dialog of roughly the same meaning using 有 (not standard!!!!)
A:"你有看过这本书吗?“ (Have you ever read this book?)
B: “我有看过。” (I have.)
Again, these examples with 有 are not considered "good grammar" if you want to score high on a Chinese test, but is frequently used among some native speakers anyway.