The terms of a contract that has been reduced to writing are definitive. If it says August then August is what you contractually agreed irrespective of prior communication or discussion. The law presumes that parties will adopt many positions during negotiations, however, when they come to actually writing down what they agreed, that written record is definitive.
There are exceptions where the mistake is obvious and makes the agreement absurd. In such circumstances if one party refuses a correction a court might order it. However, a date that is wrong by a month doesn’t qualify. Even one that is wrong by years might not - there are leases that run for decades and a court would need to be satisfies that this isn’t one of those.
Unless both parties agree that it is a drafting mistake it is not subject to correction. Of course, both parties can agree to vary the contract notwithstanding if they agree it was a mistake.