The press-up is a really common exercise that targets many different areas of the body if done correctly. However, for many, the technique is often poor. This may relate to a lack of strength or capability to perform the exercise in which case the person must modify the press up to be able to perform it better. But it may simply relate to the lack of awareness of the body position when performing the press up and this can be simply addressed using cues.
In the video below I have given 2 helpful cues to help you think about your body position. Notice none of them involve tightening the core muscles.