I have to take a makeup exam in English next week.

Literal

Next-week English [possessive-の] makeup-exam [object-を] take-not-must.

追試験 is a compound noun: 追 ('follow-up, supplementary') + 試験 ('exam') = 'makeup exam / retake.' Common in Japanese academic vocabulary. 受ける ('take [an exam]') parallels 手術を受ける.