- It was with great disappointment that Robin learnt that his cousin would not be coming.
- It was with great disappointment when Robin learnt that his cousin would not be coming.
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These two sentences are it-cleft sentences.
"That" is the most universal relative pronoun in it-cleft sentences and the sentence is correct.
You can use "when" instead of "that", but only with noun phrases when they express time. "with great disappointment" doesn't refer to time and the sentence is incorrect.
Your 1st sentence is correct because in this case that is used:—
to introduce a clause that completes or explains the meaning of a previous noun or adjective or of the pronoun it.
"that" is connecting the emotion which has been expressed in the first part of the sentence with the cause of that emotion which has been explained in the second part of the same sentence .