How should I understand 何回目か in this sentence?

Question

One assignment for my Japanese class is this Japanese elementary reader, which contains the story of Kenta and his dog, Koro. They are playing catch with a ball, and the previous sentence I translated as:

In the field, whenever Kenta throws the ball, Koro happily runs and catches it.

Now, the sentence in question is the next one:

何回目かにボールを投げたときのことです。

I get the basic idea on how to break this down:

• 何回目か is the amount of times in a sequential order, made as a question clause with か; this entire clause is targeted by the verb [clause] with に.

• ボールを投げた時のことです is "when the ball is thrown [general event]." The last part makes the "when" clause a relative clause able to be ended with a state-of-being (です). the former portion is just a clause using when with とき.

However, the use of 何回目かに confuses me, as I have not a clue how to relate it to the second part of the sentence. How should I understand its usage? Why is the 目 relevant to add? How should I comprehend the sentence as a whole, since 何回目か is the target, and an important part of the sentence?

Answer 1

何回 = how many times

As usual, adding か makes us a noun with some uncertainty:

何回か = several times

Then we add 目 (I don't know why it comes before か) to get an ordinal expression:

何回目か = the several-th time = the umpteenth time.

Altogether we have:

何回目かにボールを投げたときのことです。
It was when he had thrown the ball for the umpteenth time. ...

I'm guessing that in the next line Koro doesn't fetch the ball for some reason.