Boro host Evo-Stik Premier side Ilkeston in the semi-final of the Doodson Sports Trophy, with the winners heading through to a final showdown at Stockport County's Edgeley Park.
Kittrick has turned around the form of the club in recent weeks, pushing them away from the Evo-Stik North's relegation zone and guiding them through to the last four of the Doodson Sports Trophy.
He said: "I just want to invite Scarborough football fans down to come and support us. We'd love to see as many there as possible.
"We have a great opportunity to get into a final and win a bit of silverware.
"It has been a turbulent season and we want to get something out of this game and this competition for the supporters.
"It is about putting a smile on the people of Scarborough's faces."