Can you block consistent bad reviews from one customer? Especially one that you have made attempts to rectify the customer’s experience. Is it a good idea to just block the customer after a period of time?

On most review sites, you can’t “block” reviews.
(Although Google recently began allowing businesses to “flag” suspicious reviews, but they’re not the norm.) In most cases, you just want to respond to the bad reviewer (politely) and tolerate it. Ask them to call you to take the dispute offline. You will then want to create a strategy to get more GOOD reviews from happy customers which will eventually push down the negative reviews.

It’s generally not a good idea to delete reviews -- even if you have the ability to do so. Most people can see the true complainer for who they are.