Your advantage based on the AGPL license is, that you as the original author are free to also provide (closed source) licenses other than AGPL, which allow your potential customers to run services based on your software without having to disclose their source code. Others who use your AGPL-ed software (or its derivative works) are unable to offer this due to the copy-left features of AGPL.
Let's now look at the impact of patents.
The AGPL License in Section 11 states that every copyright holder (this includes you as the original author) is a 'contributor'
A "contributor" is a copyright holder who authorizes use under this
License of the Program or a work on which the Program is based.
and that each contributor licenses their essential patents to all downstream users
Each contributor grants you a non-exclusive, worldwide, royalty-free
patent license under the contributor's essential patent claims, to
make, use, sell, offer for sale, import and otherwise run, modify and
propagate the contents of its contributor version.
This means that all users of your AGPL-ed software, including downstream derivatives that you distribute, are licensed under your patents. But this obviously only is valid for your own code and its derivatives (including modifications contributed by others that would infringe your patents), the patent license is not valid for software with the same functionality that was independently developed from (not a derivative of) your code, a so-called clean room design. (see also here )
Once your patent application is allowed and a patent is granted you can enforce it and litigate against those that sell and/or use this independently developed software on the basis of patent infringement.
But you need to understand that a patent is only valid in one jurisdiction, there is no 'international patent'. So if you want patent protection in many jurisdictions you will have to file patent applications in all of them. And it is important to know that the patent applications themselves are basically in public domain, they don't have copyright in many jurisdictions. Therefor, by filing a patent application in a handful of jurisdiction you are publishing the 'recipe' of your 'secret sauce' in all jurisdictions free to use for everyone (only restricted if and when a patent is granted in that jurisdiction).
As a consequence, if you file your patent application for example at the USPTO, someone in Canada or Mexico might read the patent application and implement a new software based on it and offer it on servers located in their home country, and there is nothing you can do about it. If you then also file applications (based on your priority application) in Canada and Mexico, the other software provider might just move their servers to another country nearby, and there is nothing you can do about it.
Even if your process is really innovative you should not be too confident that your patent application will be allowed. In the last 10-15 years it has become more and more difficult to get software patents, in some jurisdictions it is almost impossible. You should therefore seek guidance from a seasoned patent attorney.
Summary:
Filing a patent application can be a double-sided sword. You can get some protection in some jurisdictions, but you are also opening up everything for copycats in jurisdictions where you have not filed a patent application, or where your patent applications have been rejected by the patent office.
You should carefully consider the trade-offs between relying on the protection AGPL gives you vs. the additional protections and risks gained by filing patent applications.