If i add tap water back into the tank after I drain some out will that mess up the cycle or cause a tank not to cycle?

Or you just talking about doing a water change or are you describing something more specific? No, a water change won't interfere with the cycle. You're not removing the beneficial bacteria ('BB') - they live in the filter, on the rocks, on the wall, etc, but not really in the water.

The only way I could see it being a problem is if you changed the water so much that you never allowed even modest amounts of ammonia and nitrite to build. Too much of course is harmful for your fish, but some is needed to attract/feed the BB.

P.S. Of course, never put water directly from the tap into a tank. Allow time for the chlorine to evaporate out (~24 hours), or use a chlorine/chloramine removing conditioner. And if you have chloramine in your tap, definitely go with the conditioner b/c 24 won't nearly cut it.

Thats what I meant putting water directly back into the tank. I let it sit out of course just wondering if I can add tap water back in if I use conditioner. To many containers sitting around.

If you pretreat the water with a decholrinator that removes chlorine and chloramines its fine API do a water conditioner that does both. As thats way I do my weekly water changes. The API one has instant effect and I have had no problems doing it this way.

I use prime to treat my water I was gonna start putting water directly into the tank soon cause of the lack of space i'm gonna have at the new place.

Thats fine long as you add it to water before putting it into tank is fine