Welcome to Spiceworks and its community, a community of IT professionals for IT professionals with a focus on SME. And please don't forget to read the recommendations of our field guides, especially those on getting started and on pos(t)ing good questions, including the helpful references found at the bottom of its web page.
I use the Login with Google option on the web site but there isn't an option for that on the mobile app
That's correct. I don't see such an option neither for the current Android edition of the Spiceworks mobile app to access Spiceworks Cloud Help Desk (CHD).
Can a login with Google option be added to the mobile app?
I don't know of such an option. You may vote up existing feature requests or create your own or both. For creating your own feature requests, you may need to upgrade your community privileges to next level. The getting started field guide referenced above offers you two options to do so within a few hours. The ranking and comments of feature requests is used by Spiceworks to determine development priorities when to schedule addition which feature.
You may look into this existing feature request to which extent it overlaps with your desire. I did not read all the comments to conclude to which interface of CHD this request relates or is limited. So it makes sense to add a comment that you want such a feature also for the mobile app interface. But I recommend to create an additional feature request with the scope of the mobile app directly and only. There exist two supported editions, one per operating system (Android, iOS). So if you like it just for one operating system, then please mention your preferred operating system.
when I try to manually use my email address and password I get the error message "Something went wrong"
Then please lookup your Spiceworks community account and which email address is associated. I don't know if you used the correct combination of email address and password. It should work. I tried it successfully yesterday. Perhaps your issue is temporary, e.g. due to heavy load on some back-end systems of the public Spiceworks Cloud. It does not scale well. So trying repeatedly might help to await a time when the back-end is less busy. Your error message sounds like such a temporary error. If it persists, you should contact Spiceworks support via email (email@example.com) with further details. As far as I noticed, there have been three updates on the Spiceworks Cloud yesterday. If you tried during one of those updates, it would explain your symptom and should work now. Another cause could be some scheduled maintenance with service interruption. But I'm not aware of any such intended maintenance actions for this week.