1999 Audi A4 Can not restart immediate after using lights a

I have a 99 A4 Quattro 2.8. It runs well but regardless the temperature, hot or cold, if the air/heat and light are on for a while, it won't restart right after a short power off. I have to floor the gas while turning the ignition key to get it started. And another problem is that once it starts, the acceleration kicks in slowly when flooring the pedal. It seems like it needs to drive a min or so before the full acceleration power is restored. What is wrong with it? And what needs to be fixed or replace? Thanks.