Prosím Vás je možné vynechat „that“ v následujících Cleft sentences:

It is because I have got so much work to do (that) I can't come.
What annoyed me was (that) she didn't apologise.

Jsou nějaká pravidla, kdy to vynechat nejde. Napadá mě třeba, když jde o podmět:
It was Erika that wrote that poem.

Moc děkuji

Napadá mě třeba, když jde o podmět …

Ano, právě o to jde.

It -cleft sentences

The clause which follows the it-clause is connected using that and it contains information that is already understood. We often omit that in informal situations when it is the object of the verb … (cambridge)

Your examples:

• 1. It is because I have got so much work to do that I can't come. You can't omit „that“, because „I have got so much work to do“ is not the (grammatical) object of „I can't come“.
• 2. What annoyed me was that she didn't apologise. You can't omit „that“, because „what annoyed me“ is not the (grammatical) object of „she didn't apologise“.
• 3. It was Erika that wrote that poem. You can't omit „that“, because „Erika“ is not the (grammatical) object of „wrote“.

More examples:

• 4. It was your mother who/that phoned me" You can't omit „that“, because „your mother“ is not the (grammatical) object of „phoned me“.
• 5. It was your mother (who/that) I phoned. You can omit „who/that“ in informal style because „your mother“ is the (grammatical) object of „I phoned“.
• 6. It was my sister that/who was talking to me. You can't omit „that/who“ because „my sister“ is not the (grammatical) object of „talking to“.
• 7. It was my sister (that/who) I was talking to. You can omit „who/that“ in informal style because „my sister“ is the (grammatical) object of „I was talking to“.

Moc děkuji za odpověď. Takže v následující větě by to také nešlo vynechat:

It’s in Brighton (that) Jenny has bought a new house.

Zdálo se mi, že mi to zní dobře bez that. Děkuji

… v následující větě by to také nešlo vynechat: It’s in Brighton that Jenny has bought a new house.

That's right: you can't omit „that“ in this sentence because the grammatical object of „bought“ is „a new house“.

Compare:

• It's a house (that) Jenny's [= Jenny has] bought in Brighton, not a flat. Here you can omit „that“ in informal style because „house“ is the grammatical object of „bought“.