There are two separate issues which must be addressed in your question.

One more point worth mentioning is that the HPLC system should be setup to run proper column and system flush methods after each analysis as routine. These wash methods help to reduce column and system fouling and help in reducing any build-up of material inside the system, esp after it has been shutdown (flow stopped).

It is very crucial to inject clean samples into the MS system. As you mentioned salts and proteins from biological samples and other endogens are bulky contaminants that cause the ESI capillary fouling and this can't permit samples to pass and nebulize efficiently. Thereby sample preparation & clean-up must be optimized very well.

When your capillary clogged, thread a nut and ferrule to peek covered end and screw to a valve which your LC flow pass. Purge the system with MeOH and set the flow to a low amount (50-100 ul/min. for instance). Unscrew the LC column. Start the LC flow and during Meoh flow through ESI capillary, apply heat to it using a lighter. Perform it by moving the lighter from end to end dynamically through the capillary several times and in short intervals. Do not hold the lighter to a single point of the capillary. In accordance with the application, increase the flow rate while keeping the pressure in mind. Thermal expansion and head pressure will push the fouling out of the needle.

Just try if nothing works...

We have Bruker LCMS, where the ESI-needle tends to clog. We just put it in a beaker full of MeOH and put it in an ultrasonic bath for 30 minutes. It will work.