Flight Instructor reported an airborne conflict during departure climb with a government UAS that was descending toward the airport. The flight instructor took evasive action.

Narrative

Location: ZZZ -- departure Runway XX; routing direct ZZZ VOR Date/Time: ~XA45 local Other aircraft: MQ-9 drone (border mission; IFR arrival into ZZZ) Conditions: VFR; TFR active ('HOT')What HappenedWe departed Runway XX at ZZZ on a training flight. My student read back our clearance to takeoff and turn left direct to the ZZZ VOR. Almost immediately after liftoff; tower asked if we were already in our left turn. I understood this was because a border drone (MQ-9) was descending into the area on an IFR plan; and tower didn't want us turning into the hot TFR corner where the drone was operating.During climbout; my student ran the climb checklist and did not initiate any turn below 400 feet AGL. Upon that altitude; we reported climbing direct to the VOR for the VOR XY procedure. Passing about 2;600 MSL; I saw the MQ-9 off my right wing descending out of the northwest corner of the TFR toward what looked like a 2-3 mile downwind entry from the center of the airport. My student was under the hood; so I had sole responsibility for see-and-avoid.Tower and the drone operator were trying to coordinate on frequency; and at one point the controller issued 'Right turn immediately' mistakenly using our callsign meant for the drone. If I had complied with that; we would have turned directly into the drone. Before this even occurred; I took the airplane from my student; told Tower I was maintaining my present course; and continued climbing.Our traffic system gave an alert; and based on ADS-B; I estimate lateral separation was less than 1 mile and less than 500 of vertical separation. I couldn't tell the altitude difference it happened so fast. Eventually the drone widened its downwind and we passed without incident.Why It Was HazardousThe drone was descending directly toward the airport traffic area while we were climbing out.Tower and the drone operator were focused on canceling IFR; not on ensuring separation from us.The incorrect call sign by Tower to my aircraft with the command 'Right turn immediately' created a moment where compliance could have led to a collision.I never received a clear traffic advisory; maintain VFR; or do you have traffic in sight about the drone's position or intentions.What HelpedI did kept continuous visual contact with the drone and backed that up with ADS-B monitoring.I immediately took control from my student when the situation developed.I maintained heading and climb rather than reacting to ambiguous instructions.SuggestionsBetter pre-coordination when UAS are operating IFR into a towered field with a hot TFR.Clearer; more specific traffic advisories for manned aircraft in conflict with drones.Controllers should take extra care with callsigns and phraseology when multiple aircraft are on frequency.Consider procedures so extended ATC/UAS coordination does not congest tower frequency during critical phases of flight.Move the TFRClosingThis all unfolded in under three minutes; but it created a genuine near-midair hazard. The primary concern is preventing a repeat of ambiguous instructions and lack of clear separation when unmanned aircraft mix with manned traffic so close to the airport. Something needs to be done with this TFR with the amount of training at this airport.