Hayward’s heroics, plus a fourth try from inside centre Matt Piper, helped them secure a good away victory despite fielding a mixed team which included a number of players from their second team, the Kestrels.
There was a similar story for Sheffield Tigers, who fielded essentially their second XV for their Yorkshire Cup tie at Driffield.
They ran out 27-17 winners, with captain and winger Pat Searle scoring a hat-trick of tries and others came from hooker William Burch and winger Cas Abaka. Full-back Jordan Hamilton added a conversion.
Cleckheaton will travel to National League Wharfedale for their quarter-final on February 23, while Tigers are at home against Sandal on the same day.
Sheffield and Ilkley reached the quarter-finals after Hull and Bradford and Bingley handed them the ties.
Old Rishworthians lost 33-24 at home against Novocastrians as their RFU Intermediate Cup campaign began badly.
A red card for a high tackle after just 30 minutes left the Yorkshire side with an uphill struggle, although they were only 19-17 behind at half-time.
Despite tries from scrum-half Chris Cullimore, prop Phil Kershaw and winger Doug Heseltine they could not secure the win.
Northallerton were 36-0 winners over Whitley Bay Rockcliff in the RFU Senior Vase, while Thornensians produced an excellent display in the RFU Junior Vase, beating Blyth 63-10.
The Yorkshire side started strongly, crashing over twice within the first 15 minutes.
James Fenwick and Rob Brown scored two tries each while others from Tom Holt, Scott Berry, Tom Ambler, Zak Holgate, Lee Linsdell and Oliver Cox were enough to seal the convincing win.
Moortown reached the semi-finals of the Yorkshire Shield, winning 34-32 at Beverley.
Hullensians were 17-12 winners at Dinnington in round one of the Yorkshire Shield, while Middlesbrough were handed a walkover by Selby.
Heath reached the semi-finals of the Yorkshire Shield thanks to a 36-12 win at Keighley.
Freddy Howarth and Nathan Harding both scored two to help North Ribblesdale to a 46-17 victory over Old Otliensians and reach the semi-final of the Yorkshire Silver Trophy. Other tries for the home side came from Mike Thwaites, Matt Spears and John Padley.
Wetherby ran in three tries on their way to a 21-12 win over Ripon in their Trophy quarter-final, while Yarnbury reached the semi-finals as Barnsley pulled out due to a lack of sufficiently trained front-row players.
In the Yorkshire Vase, Wibsey were emphatic 46-24 winners over visitors Ossett, while Wensleydale enjoyed a 31-0 victory at Aireborough in round two.
Halifax Vandals ran out 34-24 winners over Baildon in an entertaining Vase tie, and Knottingley won 26-14 at Harrogate Pythons.
Malton & Norton continued their North One East promotion push by scoring five tries in a 31-14 win at Consett.