What is the ruling on the birth control injection in this case, and must she make up prayers and fasting due to continuous bleeding for 16 days, and what are the general implications of this?

It is permissible for a woman to use temporary contraception if her health cannot endure pregnancy. In the case of bleeding, if a woman is accustomed to menstruating on specific days, she should not pray or fast during those days. Afterwards, she should perform ghusl (definition), then pray and fast. She must wash her private parts, bind them, and perform ablution before each prayer. She may combine prayers to ease the burden on her. Her prayers during menstruation are invalid and do not need to be made up, but her fasting during menstruation is invalid and must be made up.