the difference of having a relative pronoun

Question

He is going to cut the part of hair (that) he dyed purple a few months ago in a barber shop.

I bracketed that cause it seems like placing that is optional,
if I omit the relative pronoun in this sentence ,
does he dyed purple a few months ago still act as an relative clause?

If we have that here, I can easily understand that:
He dyed part of his hair purple in a barber shop, and he is going to cut it,cause the PP in a barber shop belongs to the relative clause.

But if we don't have that in the sentence, will the meaning of the sentence slightly change? By which I mean would another interpretation appear without the presence of that?

Another interpretation could be: Part of his hair is dyed purple a few months ago, and he is going to cut it in a barber shop?

Answer 1

A jounalist told me just a few months ago when reading something I'd written that 'that' can be ommitted about 75% of the time where it is used. It had never occurred to me before, but she was right.

So yes, 'that' is optional. And if a word is optional, in many circumstances it is better to omit it.

Answer 2

You can omit 'that' here. Because a relative pronoun is optional when it modifies the object of the verb of the relative clause. Here, 'that' (i.e., 'hair') is the object of the verb 'dyed'.

Therefore, 'that' is optional in your sentence, and it means : 'He dyed his hair purple a few months ago in a barber shop, and he is going to cut the part of it.