Is it with a fuel pump or gravity feed?
Is this a magnetto coil or a 12 volt coil?
A low battery could cause the 12 V coil not to work. And, you might be hearing the engine die, restart..
When it surges like that, I would say lack of fuel or governor.
Make sure governor arm is snug on shaft. To adjust governor, I look at an internal diagram of engine, make sure the piston for the governor is down all the way, open butterfly on carb all the way and that is the adjustment I use. Briggs and Tecumseh engines are cut and dry. The other brands are finicky when adjusting the governor so remember the setting you had beforehand.
The lack of fuel could be caused by a bad fuel filter, clogged fuel shut-off valve, clogged fuel lines, or clogged needle seat. Water ran thru the fuel filter will clog the paper filter.
You could try testing for lack of fuel by using choke to stabilize the engine or hold finger over carb throat and adjusting the air that it is allowed to have.
I have a briggs engine that I think the governor assembly is broken. Not sure how to know without looking inside the engine.
Also, try using a new sparkplug. Water will ruin a sparkplug too. If it is a points ignition, someone has mentioned that sometimes you need to set points differently to make the timing work out. Could be from a worn out points push rod.
Hopefully someone will know the correct holes the linkages go into. If it is anything like my 8 hp, you have 4 choices.