"further explanation" is correct and usable in written English. You can use it when you want to provide more detail or clarity about a specific topic. For example: "To understand the complexity of the situation, you need further explanation of the policies in place."

"further explain" is a valid phrase in written English. It is used when you want to add additional details or examples to clarify an idea. For example, "The new policy has caused confusion, so please further explain what is expected of each department."

"to further explain" is correct and usable in written English. You can use it as a way of introducing additional explanation or clarification when you have already provided some information on a topic. For example, "The atmosphere in the school was tense.