You need to manually check the oil pressure with a gauge. Of course you have to find the sending unit to do this, and maybe it'll just be a broken wire or bad sending unit.

If the engine oil pressure light comes on while the engine is running at operating speed (not idle) then you should not run the engine. First thing to do is check the oil level when the engine is cold and on a level surface. Next thing would be to check the oil filter for tightness and leaks. Next thing to do is checck the pressure sending unit on the motor to see if it is leaking. There are several other areas to check but to be honest, unless you are comfortable working around engines and hanve the tools I would turn it over to a lawn mower shop for a look around.

i got a big oil use issue with a Kohler twin SP27 using a lot of oil and fowled plug on the the oil cap side of the engine. FYI, i read somewhere on the net to check the reed valve- whatever that is- a busted or defective one leads to excessive oil use apparently. Anyway, my low oil pressure light was not working from the start, and i ran mine with low or no oil - real smart i am. Now, i am looking for a short block to get me going again. the thing runs but uses an unsustainable amount of oil because the oil rings on the piston and bore are probably damaged - i think. As i have to add oil again after a couple of hours hard running, and even one cylinder stopped firing because of the oil on the plug. I think the compression and oil rings are gone, and it would have been due to running it on a hot day/week with low oil, from what i am recollecting about the machine usage and engine noises. the machine was showing 20% pressure loss on one side, after only 200 hours. Another veteran mechanic said that all air cooled engines all use some oil as a part of normal running, and that it is important to keep an eye on your oil levels at all times, or you will get what i got. The mower repair guys said i 'dusted' the engine by not maintaining clean air filters, but i am pretty certain i damaged the engine rings and bore because of low or no oil. i am going to order a new engine from the USA , and have the old one rebuilt in a engineering shop that specializes in cylinder sleeves and pistons. Its an expensive pain in the rear but, having a spare good engine block ready to go is the only way i can think of to control the engine reliability situation long term .

When the oil light comes on, you have to shut the engine down immediately to prevent damage. If you did run it very long (like more than 30 seconds) with the oil light on, it's a good possibility you did damage the engine. Sadly, putting the oil in now is like closing the barn door after the horse has already run off. Since you only bought it a month ago, if you bought it from a dealer, you could contact them. Possibly you have some recourse since you may be able to make the case that it was low on oil when you bought it. But be nice about it, because they also have a pretty good argument that it is the owner's responsibility to check the oil. If they offer to split the repair cost with you or give you a discount on some other vehicle, they're being very accommodating.