You can cut some leaves without hurting the plant but I have learned my lesson not to over prune anything and take only a few leaves at a time and then wait a day or two and look at the plant and decide to whether to take a few more. If you take off too many leaves especially in summer and there are fruit in the darkness under the leaves, they get too much sun and sunscald and most plants do not take kindly to having too many leaves being taken off at one time. If you are training the cucumber vines up, you can remove the suckers, that usually is enough to thin them.
Happy gardening in Hawaii. Gardens are where people grow.