It is a challenging question as, it is, at least for me, hypothetical. I do not know what I would do if I were placed in that situation. Not now anyway.
The assumption in this question is that the partner made an indiscretion while he is in a committed relationship with you and is asking for forgiveness. This is the premise by which I would answer the question.
Did he come to you and told you of the indiscretion or did you find out from someone else? It makes a world of difference.
If he came to me and told me of it then the first question I would ask is "Do you wish to stay with me?"
The second question is "Do you think you will be happier if you were free of me/our relationship?"
To forgive, one must first blame. I have removed blame from my vocabulary so there is no need for forgiveness, on anything.
I would hope that the person I chose to be with with respects me enough to tell me he is in love with someone else and therefore wants out, in which case, he is free to leave at any time. I do not wish to limit the freedom of another by insisting that he stays against his will.
There may be other factors involved as you said, children, parents, pressure from friends, and in this case you will have to consider but you will still have to do what is best for you.
One happy person and one unhappy person do not make a relationship and it can be continued for the sake of appearances but I would ask myself why. This you will have to discuss wit your partner.
On the other hand, if it did happen, there is a breakdown in communication somewhere and the indiscretion, is the result, not the cause.
I am afraid my answer is limited but that is all I can think of at the moment.