Is it permissible for a woman to pray if she has taken a birth control injection and has been bleeding for twenty days, knowing that her usual menstrual period is only 8 days?

If the bleeding continues for more than 15 days, the woman is considered to be suffering from استحاضة (prolonged non-menstrual bleeding). If she has a customary period, she acts according to it. Otherwise, she differentiates (between blood types), and if that is not possible, then she relies on estimation, observing six or seven days of the month. Since your customary period is 8 days, you should observe it, and consider any additional bleeding as استحاضة. This is the view of the Hanbalis, who prioritize custom over differentiation.