This is an example of how to mate using a Hawk in endgames with King + Hawk vs king.
Not an easy task, but with the right Hawk “dance” you will prevail.
Here is the starting position:
First, you have to reduce the space for the opponent King.
For example you can proceed like this.
1.Ke2 Ke7 2. Kd3 Kf7 3. Ke4 Ke7 4. Hc4 Kd6 5. Kd4 Kd7 6. Ke5 Ke7
When the opponent king is confined in the 7th line, the Hawk “dance” begins. A precise maneuvering will bring the Black’s King in the 8th line the Hawk will checkmate.
This is a typical position and maneuvering to limit the opponent’s King mobility and push it to the 8th line.
7. Ha4 Kf7 8. Hd7+ Ke7
The Hawk cuts the path for the Black’s King, obliged to retreat to the 8th line.
9… Ke8 10. Kd6 Kd8 11. Hh5 Kc8 12. Kc6 Kd8 13. He5 Kc8
The Hawk dance begins to push the Black’s King to the corner !
14. Hb5 Kd8 15. Kd6 Kc8 16. Hd5 Kb8 17. Kc6 Kc8
18. Ha5 Kb8 19. Hc5 Ka8 20. Kb6 Kb8 21. Hf8 Ka8 22. Hd8 mate
In these endings, you must be very careful and play precise moves with your King and Hawk to push the Black’s King in the 8th’s line to mate. The maneuvering is long and could rapidly reach the 50th move regulation in case you are distracted.