Mesne profits is a funny old phrase, and a bit confusing.
First of all, pronunciation. You say meene (rhyming with clean), not mez nay.
But what are mesne profits? Well its a way of avoiding the rule that if you accept rent it implies that there is, or creates, a tenancy, and is most commonly used for payments made after a court order for possession has been made against the tenant.
This payment is often now referred to as an occupation payment or charge.
The other important use, is where something has happened to end a tenancy, for example in a common law tenancy the tenant has had a notice to quit served on him, or has done something which will entitle the landlord to forfeit the tenancy.
The landlord will not want to prejudice his right to recover possession in those circumstances, so the tenant will be told that any payments received will be accepted as mesne profits, and should not be taken as an intention to create a new tenancy.