A - Elements of good punch
Hit with your shoulder not hand (focus on the shoulder and not the peripherals).
Hips turning in the direction of the attack
Rear leg pushing hips in the direction of the attack (on ball of the foot)
Position of front leg should be enough to hold you in place and not falling forwards or becoming stuck in place. As a general rule: let your feet spread - to the front - one shoulder or a little more at first, once you get used to it, go wider and lower till you get all hitting stances in the comfort zone, then you can learn how to slide into the jump.
Aiming behind the target
Releasing the punch at the end of the rotation, this is like
stretching an elastic band to full extend and then releasing.Result: maximal speed in minimum distance and spent energy - this is a result of relaxed and controlled practice
B- Mistakes (some not all)
- Hitting with hands not using the shoulders or even the whole body weight
Having a centre of gravity leaning either to the front or the back
It should be to the middle, however, all of your body weight should go behind the punch)
So no leaning forwards (don't have your head as first point of contact after your hand let it be within the same vertical axe as your body, always front foot should be ahead of head) and no leaning backwards (you'll fall at contact).
Hips should be horizontal if watched from the front
Too much tension in the hands, only clench your fist tight at impact - which should be at best on 75% of your full extension if a straight punch or 75% of the path any other type of punch is taking (75% is your where you fist will be travelling at highest speed).
Let your punch drag you (it should be controlled with the shoulders and hips)
- of course useless movement before and after and not protecting yourself with the other hand or exposing the head or the body to dangerous openings.
C - Steps
Beginner - feel the hips
- Have your feet spread about a shoulder space
Push with the ball of your rear foot forward - direction of the punch - while at the same time bending a little bit the front foot (just a little)
- Keep head straight (chin a little down)
- Hands in place (you should be able to move your lower body with little to no effect on your hands except for rotation - this does not mean being stiff)
Once you complete your leg push (you might notice that your hips will be moving forwards) get to feel your hips and continue from there with the hips
Practice having a single motion starting from foot to hip
Intermediate - feeling shoulder
Start with steps for beginner now just add a simulated punch with the shoulder i.e. hit as if your are hitting with your shoulder (keep hands relaxed in your guard and just attack with your shoulder like the boxers do when they warm up - very small hand movement)
Practice until you have a fluid movement from foot to shoulder and you can feel the weight coming from the lower body to the shoulder and the shoulder getting more and more loose in motion
Advanced - Full punch
- Start with the previous and then just add the punch to it (Keep your focus on hitting with shoulder here and not hands)
Stretches and notes
Warm up (any type you like)
Make sure to have your shoulder fully warm up(the most basic one is a circular motion of the hands) until they are a little sore (tired) this is best.
Do some static and dynamic stretching
Systematically advance
Practice slowly and then pick up the pace
Start with one punch drills then add up
Practice on the bag (don't push, hit, and withdraw your hand directly when you punch don't let it sink in the bag)
Try to train the weak hand (your case) in other scenarios like brushing teeth, driving, etc..
Examples of Bad Shoulder forms

He is by coincidence the former many times Kata (WKF system) Karate world champ - notice that the punch looks as it could go much further and is stuck because of a stiff not extended shoulder.

Better form

Other examples

This is good form, however, notice the little lean forwards (this is the effect of tucking in the chin)
This One is a great Example
This image is from this original page.
As as side note, in Karate (and probably most old martial arts) there is a misunderstanding regarding the pulling hand, (the one that rests on the hips) and this is due to bad interpretations of forms, in fact the most simple explanation of this hand's position is holding (or locking in) the opponent's assaulting hand (and somehow turning it to gain more control) and attacking with the punching hand, however; explanations differ and vary depending on the situation and the location in a certain sequence of movements and of course the analytical perspective of the practitioner.