Lets consider it case by case:
Case 1: Charge particle is at rest. It has electric field around it. No problem. That is its property.
Case 2: Charge particle started moving (its accelerating). We were told that it starts radiating EM radiation? Why? What happened to it? What made it do this?
Follow up question: Suppose a charge particle is placed in uniform electric field, it accelerates because of electric force it experiences. Then work done by the electric field should not be equal to change in its kinetic energy right? It should be equal to change in K.E + energy it has radiated in the form of EM Waves. right? But then, why don't we take energy radiated into consideration when solving problem (I'm tutoring grade 12 students. I never encountered a problem in which energy radiated is considered.)