Goalkeeper Jessup (pictured), formerly with Bostik League North side Norwich United, joins the Clarets having completed a successful trial with Rod Stringer's side.
The tall keeper whose other former clubs include Great Yarmouth Town and Lowestoft Town, appeared in each of City's pre-season matches, as well as completing a full set of 90 minutes against Thurrock and Bishop's Stortford.
Also making the move to the Melbourne Community Stadium is 23-year-old Jonny Giles.
The young winger spent the second half of last season on-loan at Aldershot Town in the National League from Oxford United.
Giles originally joined Oxford after impressing on-trial after completing a Degree in Physics and went on to make nine appearances for the Shots as they reached the play-offs.
Speaking after completing the deal for Giles, Stringer said: "Jonny has a lot of quality top-end.
"I think he'll be a good player for us and it gives us a lot of competitions. It makes us a bit more of a threat out wide.
"I'm sure he'll be a good addition for us but, like every other playr, he has to prove himself to get that shirt".