Car hesitates and stalls, sound of air.

Car hesitates and stalls,
sound of air beating in the air intake,
exhaust comes out air intake right after stalling

I don't drive this car much anymore 1-2x per week. A few weeks ago I started stalling when starting on hills and at lights (manual transmission) which is embarrassing and not something I do.

I realized that if I punch the accelerator a bit the car stalls, I have to feather it really carefully to get it to 'catch' and rev up.

With the hood up I can hear a gentle bap-bap-bap air beating noise in the air intake when the car is idling. The frequency is about 7x per second. It is not constant, but the car is more likely to stall if I rev it when that sound is happening.
When the car stalls, exhaust comes out the air intake.

Is this a sticking valve? Or just vacuum hoses? Or one of the other 10 systems that could be involved?

Many thanks
~Tobi