CRJ-900 pilot reported a bird strike followed by a flap malfunction during approach. Flight crew performed a go-around; experienced a second bird strike; and returned to landing.

Narrative

Aircraft X. ZZZ1-ZZZ. OE Captain Upgrade. PF Check Pilot Right seat. Captain Upgrade PM Left seat. Night; VMC conditions.On approach ILS XXL at ZZZ; 2;500' MSL and flaps 8. We were outside the FAF when we encountered a bird strike on the captain side windshield. We continued and set flaps 20 and then 30 on schedule. In the transition from flaps 20 to 30 we received a FLAPS FAIL Caution message on the EICAS. We recognized it immediately and notified ATC of the issue and performed a Go Around. Their instructions were to fly runway heading a climb to 3;000'. During the Go Around the PF called for flap retraction and gear up. Once we had retracted the flaps; we realized we had done the accidentally done procedure incorrectly. When we brought the flaps from 30 to 20; the Falls Fail Caution message cleared from the EICAS. During this; we received a secondary bird strike just below the captain windshield. We completed the After Takeoff checklist. The Check Pilot transferred the flight controls and radios to me when we leveled off. The Check Pilot then began to run the QRH procedure. We reached 'select last flap position and reselect the last slat/flap position' and recognized we were in the clean configuration. We decided to slow the aircraft down and try reselecting flaps 30. Doing so; the Flap Fail caution message did not post; which the QRH led us to 'No further action required.'The Check Pilot was communicating with Maintenance at ZZZ2 through radio to use all possible resources. The Check Pilot relayed to Maintenance that we had raised the flaps to the clean configuration and we cycled the flaps and did not receive the Flaps Fail caution and that we technically completed the QRH procedure. Maintenance then asked us what our current weight was to provide us with a landing distance requirement for Flaps 1. Once we received the landing numbers; we determined that XXL had adequate distance for a Flaps 1 landing. Since Dispatch only provided Flaps 1 landing distance; we determined that was the landing that Dispatch would want us to perform for safety. I brought up the point that if we were to try and lower the flaps; that there was a potential for damaging that flaps further. The Check Pilot agreed with me and we would assume that the flaps were inoperative.We proceeded with the QRH portion. After completing the procedure; we derived at Flaps 1 landing with a VREF speed of +24 kts. We then called the flight attendants and advised them of the situation; which could then turn into an emergency. We then made a PA to the passengers advising them the reason for the go around; a flap failure. The Check Pilot described to the passengers the purpose of the flaps and that we had talked with maintenance personnel that confirmed that we will be making a landing back at ZZZ.I gave the controls back to the Check Pilot who became pilot flying and I became pilot monitoring. We requested priority handling with ATC and requested that emergency vehicles be on standby for us. ATC gave us vectors back to the airport. We had previously completed the descent and Approach checklists to ensure we were prepared. We began the ILS XXL approach and landed on the 1;000' markers at VREF on XXL. We vacated the runway and came to a complete stop on taxiway 1 for ARFF (Airport Rescue and Firefighting) to take temperature readings of our brakes. We shut down both engines; with the APU running; so they could complete the readings. Our brake temperature readouts began at 1 and slowly progressed to 3 and remained at 3. ARFF notified us that our brake temperatures were 260 degrees Fahrenheit. Once they determined our brakes were not overheated/ on fire and the aircraft had sustained no damage; we started one engine and taxied to the gate with ARFF following us.