Certified GKH Gardening Expert
This appears to be fungal or likely to be temperature or humidity related.
If it is too hot and dry, this will happen to melons. You can water it more often to help, unless the soil stays wet most of the time. If it does remain wet, then you could look to a fungal issue.
To correct this, you will need to apply DOLOMITIC lime and WETTABLE sulfur during the off season. This will kill anything in the soil as well as recondition it. You will also want to add some organic matter like compost or some liquid nutrients during this time as well. This will get you ready for the next planting.