Hey!
They did identify the link between a gene and polypeptide. They hypothesised that the mould would require certain enzymes to convert the mould into amino acids. After striking xrays to the mould that they were growing, a mutation occurred within one of the codons, and was returned in the minimal medium (culture with the minimum amount of nutrients necessary for growth) which the mould was growing and they observed whether or not it flourished or not. If it survived it was discarded and if it was dying, they changed the minimal medium discovering that it grew through the supplication of certain nutrients and amino acids. Originally it was thought that a mutation in one gene affects the development of one protein, which was later disproved as a gene only codes for polypeptides and it requires multiple polypeptides to generate proteins.
TL;DR
Relationship
mutation in one Gene=alteration in the amino acids generated=alteration in one polypeptide
mutation in one gene ≠alteration in the generation of one protein
I hope this helped