Using 'that' does not make the sentence incorrect. The sentence "I am writing to apologise for the meeting THAT I missed on October 20th about development plans for the company" is OK. You can include the relative pronoun 'that' if you wish, but it is not necessary.
You asked "When not to use that?".
Here's an answer:
You can omit the relative pronoun (that, who, which) when it refers to the OBJECT of the clause; you cannot omit the relative pronoun when it refers to the subject of the clause. For example:
"I apologise about the meeting that I missed on October 20th."
"I apologise about the meeting I missed on October 20th."
The relative clause here is "the meeting that I missed". In this clause, 'I' am the subject and 'the meeting' is the OBJECT. The relative pronoun 'that' therefore refers to the object and can be left out: it is fine to say this sentence either with or without 'that'. Native speakers tend to omit the 'that', especially in speaking and informal writing.
"I missed the meeting that was held on October 20th."
In this sentence, the relative clause is "the meeting that was held on October 20th". In this clause, the meeting is the grammatical SUBJECT, so the relative pronoun 'that' is essential. You cannot omit it.
I hope that makes sense.