The pair - who scored Rhinos’ tries in last week’s game - will miss the trip to Salford Red Devils on Sunday, April 15 and visit of Wakefield Trinity five days later.
Prior received a two-match penalty notice for Grade B tripping.
It is the second time he has been charged this year, so he was handed the maximum punishment for the offence.
Gannon was handed a similar punishment over a Grade C charge of touching an injured player following the tackle that saw both Kruise Leeming and Lachlan Coote leave the game for head injury assessments.
The 18-year-old's previous good record meant he received a two game ban, rather than the maximum three.
Ironically, Zane Tetevano and James Bentley, who play similar positions to Prior and Gannon, will be available for Rhinos' next match after each completing a two-game ban.
Salford will face Rhinos without former Leeds captain Kallum Watkins who was handed a one-game penalty notice for grade B dangerous contact in last week's defeat at St Helens.
Wakefield's Jacob Miller was also charged with grade B tripping, in last Thursday's loss to Huddersfield Giants.
He was handed a one-match penalty notice and will sit out Trinity's next game, at Toulouse.
Teammate Liam Kay avoided a ban after receiving a zero-match penalty notice for grade A dangerous contact.
Other charges were:
Albert Vete (Hull KR - grade B high tackle) - two-match penalty notice.
Matty English (Huddersfield Giants - grade B dangerous contact) - one-match penalty notice.
Oliver Wilson (Huddersfield - grade A dangerous contact) - zero-match penalty notice.
Albert Vete (Hull KR) - dangerous contact.