Ramp Tower Controller reported creating a critical ground conflict; by clearing one aircraft to push back; being distracted by a fueling issue at another gate and previously cleared another to taxi into the ramp area.

Narrative

On the PM shift of Day 0; I was working gate XX ramp controller at ZZZ airport. It was an extremely hectic day with a lot of aircraft movement; deicing; and FAA slow to clear ramp. I had just been thru a couple of rough complexes with having as many as 10 aircraft holding on the ground waiting for a gate. I cleared in a local aircraft to gate XX and prematurely showed him ON the gate on my board; I do not recall how long the time was that I cleared him in to the time he entered the alley. Some minutes later; a live trip off gate XY called for push; and I approved the push; believing I already had gate XX on the gate. As gate XY was pushing; the local aircraft for gate XX entered the east line; almost making contact with the pushing gate XY aircraft. The local aircraft called me and asked if gate XY could move back in so he could complete his taxi to gate XX. There was no contact of the 2 aircraft and no damage; but a close call.Identification: Thinking I already had gate XX parked on the gate; as I had mistakenly shown on my board; led me to clear gate XY to push. Then the local called at top of the alley asking for room to be made. I guess I'm also frustrated by the fact that the local aircraft seemingly prematurely called approaching. Although; again; the exact details are blurry. I don't recall exactly where the local aircraft was when he called approaching.Cause: The exact details of what led up to this are a blur; as it was an extremely hectic night (and time period). I did not witness the local aircraft entering the alley as gate XY was pushing; as I believe I was conversing with the fueling department in the corner of the room regarding the need for fuel on a local aircraft on gate XZ. And I think I as I was getting ready to advise the gate board of the fact of maintenance crew being on gate XZ; is when I noticed we had a major problem and that I had made a mistake; with respect to the gate XY push; and the local aircraft for gate XX. As I recall; the local; from the time it called approaching; to the time it entered the alley; took a longer than usual time to show up at the top of the alley. What happened is my fault and no one else's. However; I'm frustrated by the extraordinary workload of number of aircraft's I had to deal with last night. The high number of aircraft movement in the alley; combined with delays for deicing; combined with the FAA slow to move aircraft out of the ramp; made for a very challenging and; in my opinion; less than safe atmosphere.Response: I thought about advising management immediately; but ultimately; I did not notify anyone of the near-incident because there wasn't an incident.Suggestions: Less aircraft movement (particularly regionals). Local aircraft need to call approaching closer to gate.