Normally the gap is very small so that the effect due to the magnetic field betwen the gap can be ignored.

For a cyclotron device, we usually only interested in the final beam came out at particular radius (with particular energy/momentum).
$p=mv=qBr$
The device was developed to accelerate charge particle (due to electric field between the gap) to high energy.
The detail of the trajectory usually is not so important.

If you are really interested in it, you can assume there are the same uniform magnetic field between the gap,
and find out is it going to produce similar result.