Working on a skilled worker or Health and Care visa means that your right to remain in the UK is linked to the work you are doing for your sponsor.
If you want to change employers, your new employer will need to issue you with a new certificate of sponsorship, and you will need to apply to transfer your visa to them. You cannot start work with your new employer until this is confirmed and you are able to isse your new employer with an online share code.
You will need to apply to update your Health and Care Worker visa.
Be aware : If you have signed an employment contract with a repayment clause, you may be need to pay fees to your employer depending on the terms of the repayment clause you have signed.