Hi Alex, It looks like Pythium spp but the white protrusion is strange, can you take a small portion of the rotten cucumber tissue and put it in petri dish with water for 2 days and look for zoospores and oomycetes? You can take a picture under a microscope and post it I can help to identify the fungus.
There are many fungi that affect cucurbit plants, including many, and they are similar in their symptoms of external infection as Botrytis , Colletotrichum orbiculare and Fusarium oxysporum or downy mildew diseases. Therefore, it is better to grow fungi on a PDA medium or direct slides and the microscopic photo is best for diagnosis.