A convincing win over Carperby saw Unicorn go top of the table on Saturday.
Liam Reeves has been on fire this season and bagged his 16th after just eight minutes.
First half goals from Shayne Ketley and Clark Gunn, and a late strike from Rory Atkinson, ensured all three points for the Richmond team and meant they leap-frogged Richmond Mavericks, who were playing a cup fixture.
Richmond Town Reserves also had a comfortable win against a Wensleydale team, beating Spennithorne 5-0.
Josh Woof got a couple with Uchenna Anyadiegwu, Ronan Donoghue and Harry Beresford also netting.
The only other fixture was a tight affair involving Hawes and Leyburn.
Jack Hammond opened the scoring for Leyburn with a good finish from the edge of the box.
A bullet header from Brett Calvert from a Tommy Calvert corner saw Hawes Level.
But Leyburn secured the win when Brandon Acton tapped in from close range following a Luke Wilkinson free-kick.
Both teams had chances late on but Leyburn held onto the win.
In the quarter finals of the Medals Shield, Richmond Mavericks overcame Catterick Garrison.
Daniel Johnson scored for Mavs in the first minute with Samuel Pickersgill making it 2-0 from the penalty spot.
Catterick pulled one back through Jonathan Holmes before substitute Callum Plumb bagged his first goal of the season to secure the win.
Catterick Garrison 1-3 Richmond Mavericks
Hawes 1-2 Leyburn
Richmond Town Reserves 5-0 Spennithorne
Unicorn 4-0 Carperby
Fixtures – 19.12.20
North Riding Cup
Carperby v Hawes
Reeth v Richmond Town Reserves
Richmond Mavericks v Spennithorne
Unicorn v Catterick Garrison