2003 grand marquis

I have a 2003 grand marquis that won’t start with the air
filter connected to the throttl body. I remove the air filter
And with the throttle to the metal it starts but won’t stay
on. If for some reason stays on I hear a coughing through
the throttle body. If I press the gas several times while on
sometimes stays on but as soon as I replace the filter it
turns off. Yes it does have a clean filter. Fuel pump has
30psi when on. Sometimes it stays on perfectly with
everything connected but if I turn it off it won’t start unless
I remove the air filter. Has anyone had that problem if so
how did you fix it what was the problem