I assume that you have a typical Assured Shorthold form of tenancy agreement.
If it is not, the lease will need to clearly say so.
On the basis that it is a normal Assured Shorthold, the landlord is responsible for all repairs.
Tell the managing agent to do their job and advise their client Landlord that the property needs a repair. It will be for the Landlord to sort whether it his/her fence or a neighbour to sort and they should tell you what is going to happen to fix the fence.
Frankly, most managing agents do not understand their job very well especially if they are not qualified in any proper way. Over the years, I've come across all sorts of nonsense including the idea that a tenant cannot change the lock of the outside doors because the Landlord wants to be able to enter when it suits them.
NB. A tenant has a lease which, by law, ensures "EXCLUSIVE POSSESSION". It is simply not a lease if there is no exclusive possession granted. This means the tenant has every right to ensure no-one enters without their permission and has every right to secure the external doors with their own locks. Obviously, this is subject to rights by the landlord or agent to enter upon suitable notice to sort repairs or inspect occasionally.