What's your definition of a cheap scope? Under $200?
My recommendations, in order of price:
Weaver V9 3-9x $150ish
Bushnell Elite 3000 (closeout) $150-300
Sightron $150-350
Redfield $150-400
Swift $100-250
Simmons Aetec $130-200
I will never again throw money away on a cheap Chinese scope like the BSA's, Tascos, or entry-level Bushnells. I have wasted too much time and ammo trying to get them sighted (or re-sighted) in. I'm sick of not being able to focus on the reticle and the target simultaneously.
You say you want an adjustable objective, but many of the cheap scopes are going to exhibit parallax problems even with an AO. And on any power up to about 16, a 50mm objective is just extra weight. Especially on a cheap scope, a bigger objective doesn't necessarily mean a better image. It might be brighter but it won't be better.
I strongly recommend getting a lower powered model of a good brand. These will often have better resolution than a higher magnification Chinese scope.