We have had boxers in training who had problems with hand speed. Some of it was due to what I call laziness. They had become so acostomed to throwing slow and calculated punches that they were getting counter punches before the defensive stance and footwork could kick in.
Can you improve speed? Absolutly. Any trainer knows you can improve anythng, if you first recognize the problem, put your mindset into improving it, and then devote concentration and time into the training.

Improving hand speed is simply learning the technique of 'snap punching'. Have a video made of yourself in training, or in actual fighting, and see after you throw a punch if there is the slightest hesitation after delivery, in returning your arm and hand back into a defense posture, or a follow up punch. Chances are you will see it.

Simple solution: Work on snap punching on the heavy bag with a partner. First work on a snap punch and how fast it takes to return to your protectve stance. Then move to concentrate on delivery of a snap punch with force behind it. Next move delivery of a power snap punch and movement of the body to the side or backward. It takes time and concentration at the beginning until he becomes muscle memory. Snap punching is tied into fitness and stamina, so keep that in mind in your overall training.