Is a "that" missing here?

an adult who gave his word the boy would be supervised
The above chunk of language is from a collins dictionary. I wonder whether "that" should be added following "word" since "word" serves as the antecedent followed by an attributive clause "the boy would be supervised". What is your opinion?

This is correct.

"That" is not necessary in this case.

I think it would sound much better with "that".

I agree with furrykef. The version without "that" sounds unnatural to me, like something I'd read in an online news article. I wouldn't go so far as to say that the version without "that" is incorrect, but I don't think it would be allowable in my idiolect.

An adult who gave his penis to that boy should be murdered

I would myself also agree that the version without "that" sounds quite unnatural (and almost ungrammatical at that).

Sounds natural.