If a server program is licensed with an Open Source license, the license terms apply to the server software itself, not to your own files that you host using that server.
For the Affero GPL, your obligations are the same as for the GPL unless if you modify the server software. The relevant part of the AGPL is as follows (emphasis added):
- Remote Network Interaction; Use with the GNU General Public License.
Notwithstanding any other provision of this License, if you modify the
Program, your modified version must prominently offer all users interacting
with it remotely through a computer network (if your version supports
such interaction) an opportunity to receive the Corresponding Source
of your version by providing access to the Corresponding Source from a
network server at no charge.
I presume my code would run in the same process as the AGPL code,
hence it would make a combined program is this correct?
Hosting an executable file with a server does not make that executable file and the server program a combined work. The GPL refers to this situation as an "aggregate" (empasis added):
A compilation of a covered work with other separate and independent
works, which are not by their nature extensions of the covered work,
and which are not combined with it such as to form a larger program,
in or on a volume of a storage or distribution medium, is called an
“aggregate” if the compilation and its resulting copyright are not
used to limit the access or legal rights of the compilation's users
beyond what the individual works permit. Inclusion of a covered work
in an aggregate does not cause this License to apply to the other
parts of the aggregate.