We didn't prime walls the first time after putting up drywall and we painted it. We put two coats on it already and can still see drywall patches. Can we use paint with primer to fix this using same color for third time?
No, not well. Each coat of paint is no better than the layer below it. Primer is made to stick to things like drywall, paint is made to stick to things like primer.
Your likely result is paint that peels off in sheets like this wall, which was painted with two layers of latex paint directly on wallboard:
The drywall patch visibility is a separate matter. Drywall mud goes on top of primer just fine, you can patch then prime again. Some professionals even prefer doing it that way. If you do it yourself, know that very thin adjustments can be made (get the purple stuff so you can see where fresh mud has stuck).