Personal training should be just that. Personal. An exercise program that is designed specifically for you. The program will be geared toward your goal and to help strengthen or improve areas that are in need of that. There are so many exercises and ways of exercising these days that it is impossible to try and do them all in a weekly program. That's why focusing on what's needed and prioritising correctly are so important. I'll do all of this for you and help you to understand why it's being done that way. Your exercise should be a part of your normal health routine, all designed to optimise your physical and mental health and wellbeing.
All clients have to undergo a history check and physical assessment before proceeding to an exercise program. In some cases I may require you to get clearance from your GP before we begin but this is rare. If your assessment flags up things that will need to be corrected before we start with the program I'll let you know. Many times corrective exercise can be done as part of a program but not always. It depends on the problem.
For example if you can't get your hands overhead properly I'm not going to give you any military presses to do. However some other exercises may be possible. If you can't squat down facing a wall without headbutting it, no kettlebell swings for you I'm afraid until that's fixed! If you can't touch your toes no deadlifts etc. Giving exercises when there are underlying mobility, stability, or imbalance issues is a quick road to getting injured or at the very least developing poor and dysfunctional movement patterns. If you don't care and want to hit the exercises hard anyway well, there are other personal trainers out there. In 22 years of personal training I've only ever had one injury with a client, and that was a moderate hamstring strain during a maximal effort sprint.
The other factor to consider is that. When you exercise and are restricted in your mobility or with imbalances, it's far less enjoyable and rewarding. By taking the time to make sure you are in good shape to perform exercises, you'll make quicker progress, enjoy it more, and be able to push harder without fear of hurting yourself.