I am going to have to use it quite a lot today.
I am going to use it quite a lot today.
If both are correct, when should I use to have to use and when to use?
The construction HAVE + to VERB in your first sentence is what is called a ‘semi-modal’; it replaces the ‘full modal’ auxiliary verb must, exactly as the semi-modal BE going + to VERB replaces the full modal will.
I am going to use it = I will use it.
I have to use it = I must use it.
These semi-modals are more flexible than full modals. The full modals can/could, may/might, shall/should, will/would, must have no infinitive or participle forms, so they cannot act as the complements of other auxiliaries. For instance, you cannot say
✲ I will must use it.
✲ I was musting use it.
✲ I had musted use it.
If you want these sort of constructions you must substitute the semi-modal:
OK I will have to use it.
OK I was having to use it.
OK I had had to use it.
So your two sentences mean different things:
I am going to have to use it quite a lot today. = approximately, It will be necessary for me to use it quite a lot today.
I am going to use it quite a lot today. = I will use it quite a lot today.
Be careful to distinguish the construction HAVE + to VERB from the related construction HAVE + NP + to VERB (from which, historically, it derives).
I have this tool to use. = I have this tool (which) I am to use.
I have a job to do. = I have a job (which) I am to do.
Here HAVE is not an auxiliary but the lexical verb meaning hold, possess. The NP is the object of to VERB, and the implied subject of to VERB is the same as the subject of HAVE.
✲ marks an utterance as unacceptable
OK marks an utterance as acceptable