Is it a riding or push mower? Either way it sound's most likely you've got a fuel flow problem, if it has a filter inline on a fule hose that needs to be checked or just replace. If no filter on a hose then the filter on the carburetor itself is likely causing the clogging problem. A restricted fuel flow is why it runs better with the choke partially on. When the choke is on, that reduces the air flow, turning the choke partially on is bringing the air/fuel mixture back into balance, less fuel getting feed, less air needed to keep it running. On some small engine carbs, with the choke on, that also helps pull a little more fuel in on those little carbs.
It's a common problem with a lot of push mowers that are say, 10 or more years old with Briggs & Stratton engines, new models are different setup. If it's the newer style, try leaving the gas cap loose, some of the newer ones the vent in the cap meant to let air into the gas tank can clog and the tank creates a vacuum and once that happens it won't get the fuel flow it needs, like it running out of gas, see the video below.
On those older style B&S carburetors there is a very fine mesh screen made out of plastic that is part of a tube attached to the carb that picks the fuel up from the gas tank. The carb is screwed right onto the gas tank, that tube w/filter sticks down into the tank. It's a fairly easy job to take apart, the tube is pressed in, a few twists and it comes right off the carb. I pull the tube off, spray some of the carb cleaner into something like a skinny pill bottle and soak the tube/screen over night then spray the screen out from the inside with carburetor cleaner and it will clean it out. A visual look at that screen won't show it's clogged, it's really fine mesh like a silk stocking, but if you try to blow though it you'd find it very hard to even get air through it. Over time it will gum up from gas going bad, usually from being left in it during the winter months, that's enough to restrict the flow. If it's a push mower, is it also harder to start then when it was new? That is another symptom of the screen clogging up. I can see if I can find a post with good pics showing how to take the older style apart or maybe a youtube video showing the steps to fixing that type.
My one neighbor, the town drunk who use to be the UPS driver, well until he lost his job after his 3rd DWI, he went to work in a snowmobile shop, he tells everyone he's the "head mechanic"

He has a older style push mower that has that problem, it's been like that for the last two years, I've seen him numerous times trying to get it started and messing with it trying to figure it out, he has to use ether to get it running then it spits and sputters the whole time he's mowing, I laugh, the head mechanic can't figure it out LoL. Little does he know, the snowmobile shop he's working at, my cousin owns it.

I had a chat with my cousin, he told me he's does odd jobs around the shop and small jobs but he's no great mech, he laughed when I told him he's been telling people in town he'd the top tech.
This guy has some really good tips! I enjoy watching him too.