They will host South Yorkshire League champions Elsecar on or before April 29.
If they are successful Heaton will be at home to Papplewick and Linby CC or Treeton.
If South Yorkshire League side Treeton win it will see former Hanging Heaton favourite Steve Foster returning to Bennett Lane.
Victory at that stage would then see Heaton at home again in the Group Two semi-finals.
Among the teams they could meet at that stage are Whitley Hall, who triumphed in the Black Sheep Yorkshire Champions Trophy final at Bennett Lane last August.