If you conquer then move the unit to another battlefield, after that can you play a unit onto the first conquered battlefield?

No, you cannot play a unit onto that battlefield after moving your only unit away from it.

According to the rules and established FAQs:

1. Control Requirement: To play a unit from your hand to a battlefield, you must currently control that battlefield.
2. Loss of Control: You maintain control of a battlefield only as long as you have at least one unit present there (Rule 184.4.a). If you move your last unit out of a battlefield, you immediately lose control of it (Rule 184.4.c).
3. Ruling: As stated in RiftJudge FAQ #5737: "No, you cannot play a new unit into a battlefield if you have no units there. You need to have a unit at the battlefield and be in control of it to play additional units there."

Correct Sequence: If you wish to replace a unit at a battlefield without losing control, you must play the new unit to that battlefield first while the original unit is still present, and only then move the original unit away.