SCT Controller described a confused IFR clearance pick up event complicated by less than adequate radio and RADAR coverage.

Narrative

Aircraft X called for an IFR pickup for an approach to VNY Airport; VFR at approximately 4000 feet MSL. I advised the pilot unable due to altitude and traffic; but I advised the pilot; if able; to maintain VFR at or above 5500 MSL; an altitude above the MVA for said ILS approach and clear of Aircraft Y level at 5000 MSL enroute. Aircraft X complied with the altitude and climbed VFR to 5500; so I gave the pilot a heading towards the final and advised to maintain VFR. Once Aircraft X was diverging away from Aircraft Y; I was going to give the pilot an IFR clearance. Just as I was about to give the clearance; the pilot stated that they would have to turn westbound soon to avoid weather ahead. Prior to the pilot turning westbound; I gave a clearance to fly heading 360 and maintain 5000; however it appeared the aircraft turned West and started a descent from 5500 to an altitude to maintain VFR. I turned both aircraft further away from each other and issued traffic. I did ask the aircraft whether they were turning right to a heading of 360; and they replied affirmative; unfortunately this turn occurred after they had already started a left turn. I actually did not observe the initial westbound turn because apparently the BUR RADAR West of VNY is so bad that it drops; or loses smaller low level targets constantly. However; the pilot of Aircraft X stated that they never copied an IFR clearance; and were still VFR; so I re-issued Aircraft X an IFR clearance that was read back; and was well clear of traffic. The pilot also stated that they had a hard time reading my radio transmissions. Recommendation; outside of better radios for the BUR area; I would say the RADAR is in serious need of some type of adjustment West of VNY. We are losing far too many targets ( primary and secondary). Next time I'll certainly ask if the pilot is going to make an immediate turn prior to trying to issue an IFR clearance; and if so get a read back that is certain.