¿When and what factors affect the correct functioning of control points in the cell cycle, and how mitosis and meiosis are affected when control points fail?

the cell cycle control points are affected when there is a mutation in them, such as the p53 protein, or also called guardian of the genome, is responsible for inducing the cell's response to DNA damage, stopping the cycle cell in case of mutation, if these control points fail, the damaged cell will continue to duplicate and can cause cancer.

Gene products called cyclins  I assume, from memory, that they are proteins.  The cell is forced into apoptosis if there are genetic anomalies in the cell cycle.  I will give you more info when I do some additional reading.