Unless the OP has the mower or tiller attachments for his Ariens
Snowthrower, it does not now have, nor never has had, an air cleaner.

To the OP, there is another carburetor adjustment on the SIDE of the
carburetor (two actually) and one on the speed control. The screw just
above the edge of the bowl is the idle mixture screw and the one at the
top of the carb is the idle speed screw. The screw on the speed control
is the top governed speed. The idle mixture screw on an HS50 (HSSK50) or
an H50 (older heavier engine never marked SK) should be between 1 and
1.5 revolutions counter clockwise from lightly seated. However, NEVER
attempt this adjustment with the carb in any position other than normal
operating (upright) position. There is a check valve behind that screw
(a metal pin that floats up and down inside the carb) that will be bent
and render the carb junk if you tighten that screw down in any other
position, even lightly tightened.

Stormin Mormon wrote:
I've worked on some Briggs, and some Tecumseh. Not sure which yours has.

My suggestion. Remove the air cleaner to get more room to see. On top of
the carb, there is a somewhat triangular metal plate, that hooks to the
throttle. And typically has a stif metal rod that goes to the governor, and
to the spring that pulls the throttle open. This throttle plate should have
a small slotted screw, which sets the lowest speed for the throttle. Idle
speed, they call it. Give that small screw 180 degree turn clockwise, and
see if that helps.

While you have the machine this far apart, consider removing the carb, and
replacing the gasket between the carb and the engine. Two nuts, and then the
carb should come off easy enough. Those gaskets are known to go bad. If the
gasket comes out easy, give it a very light coat of Permatex #2b non
hardening gasket sealer, and put it back in. Sometimes air gets in past the
carb.