I'm back. The picture helps, but I have questions about it. Where are the doorways/windows, how big is the existing can and is it over the sink?
My kitchen is 10x14 and I have nine 5" cans on three switches. There's at least 3 too many cans. I wasn't consulted. My wife and son in law masterminded that job.
Here's some things to remember in doing the layout:
If you have switched lights in place, 25% of the work is already done.
It's better to have a plan and change it than to not have a plan. So, get up a ladder and put post-it notes where you want your lights. If it falls on a joist, you can move it over as little a three inches to place the fixture and shift the other lights around to match things up.
Tentatively locate the ceiling joists by removing the existing light fixture and finding wood. In a perfect world, the joists would be spaced evenly on 12 or 16 or 20 inch centers depending on several factors; regular boards, trusses, etc. You may be able to measure at the electrical box adjacent to the existing light to mark the ceiling joists all across the room. I double check the joist locations by driving a little bitty drill bit into the ceiling sheetrock right next to the crown molding, if there is any. Move your post-it notes around to stay between the joists and produce an eye-pleasing pattern, and to put light were you need it. Make something line up or square up with something somewhere in the lay-out, and it'll look nicer. I hope that made sense. If you have to relocate one can, you will have to scoot the rest of them around a bit to maintain even spacing. Try to make all placements look as if they were part of a plan.
Other considerations:
Put the cans 12" off the face of the wall cabinets. They function better if they shine on the countertop and not on the back of your head.
Dimmers are your friend.
Be aware of what's upstairs; plumbing, service cable, hvac ducting, etc, and be careful.
Poke a hole in the center of your prospective can location and stick a long screwdriver or similar rigid metal tool up in the bay and swing an arc around and feel for wood, piping and so forth.
I used to ask decorators where I should place the fixtures and they typically gave me the deer in the headlights look. They could probably tell me where they would like to put them, but didn't know where they actually could put them.
Dollar.