I tried to replace a plug-in 4 pin CFL and ended up having the socket stuck inside the can. How I can get it out? There are two clips at both sides in the can fixture - can I push these to get the broken bulb out?
That type of bulb just snaps in and out with a little pressure. It probably broke because the UV light had weakened the plastic, or because it force was applied in a sideways direction.
To remove it, turn off the light via the switch or circuit breaker. Grab the remaining plastic with a needle-nose plier and pull it straight outward. Do not twist or tilt or you risk damage to the socket.
Those clips are designed to hold the bulb in place, your fixture is designed for a 4-pin CFL. You need to either replace the bulb with an identical replacement or pull gently down on the fixture trim, remove the trim and replace the socket with one which accommodates the bulb of your choice (as long as it does not exceed the wattage and/or heat limitations of the fixture).