Learn English – What (an) exciting news it is!

The "at the time" is a red herring. "what was at the time" is not a linguistic unit. Rather, the key structure is "what was [complement of was]": "what was one of the most exciting fields".

"At the time" is just an additional modifier on "was", which must refer to some time that had been given earlier.

This is a "what was one of the most exciting fields" is a special type of clause called a wh-clause.

The canonical sentence behind the clause is "one of the most exciting fields was what". This could be used by itself as a question: "What did you say? One of the most exciting fields was what?"

But usually the wh-word (such as "what" or "who") undergoes "wh-movement", whereby it is transported to the front of the clause.

That clause is then nominalized: it serves a noun denoting whatever the wh-word refers to, and we can use it in various ways:

My goal is to increase my knowledge of what was once an exciting field. [I want to increase my knowledge of something. That something was an exciting field once. I.e. I want to increase my knowledge of a field that was once exciting.]

What you're doing is not correct. [You are doing something. That something is not correct. I.e. You are doing an incorrect thing.]

In such examples, we can replace the wh-clause by a noun, and still have something grammatical:

My goal is to increase my knowledge of physics.

The bank account balance is not correct.

Now about the sentence:

At the end, my goal was to increase my knowledge in X and what was at the time one of the most exciting fields in Y.

Firstly, "knowledge in X" should probably be "knowledge of X". "knowledge in X" is not out of the question depending on what X is. For instance we can have expertise in physics, but knowledge of physics (not in). We can be knowledgeable in physics, on the other hand.

Your "at the time" phrase must refer to a time which was given earlier, otherwise it is dangling. It raises the question "at what time?" We know that the time was the same as that time when you had the goal. I.e. you had a goal to study a field in Y. At the time when you had this goal, this field in Y was one of the most exciting fields in Y. (But we do not now what time that was; it must be given in some earlier sentences.

If your intent is to say that X was one of the most exciting fields in Y, then the sentence is wrong. It should be:

At the end, my goal was to increase my knowledge of X, which was was at the time one of the most exciting fields in Y.

Your sentence says that your goal was to increase your knowledge in X, and to increase your knowledge in some unnamed field, which was one of the exciting fields in Y. At this point, your reader or listener will expect you to reveal what field that was. A very short way to do that is to use phrase headed by the word "namely":

At the end, my goal was to increase my knowledge of X, and of what was at the time one of the most exciting fields in Y, namely Z. ["Namely Z": That field I am referring to which was one of the most exciting fields in Y was the field Z.]

The questions are all about the completed past (nothing carrying on into the present), so your answer must be, too. But you need not use the perfect; any past will do.

You had to be there

You would have had to be there

You had to have been there

You would have had to have been there

You would have to have been there

All are about the past, and so they all work.