The ministry has the power under the Immigration Act 1958 to intervene in your case when he thinks it is in the public interest. What is and is not in the public interest is for the Minister to decide.
The Minister is not legally bound to intervene or consider intervention. When a Minister intervenes to make a more favorable decision, it usually means that the Minister is issuing a visa.
However, only a small number of all requests for ministerial intervention are successful.