To troubleshoot initially;
1. Is it happening with particular dataset/s or all of them?
2. If it is particular dataset/s, try dropping cache (Manual Reset). Rebuilding cache may take some time which depends on how big/small corresponding feature service is (number of features).
3. If it is still not working try un-sharing and re-sharing the concerned item/s from Public and Open Data group.
4. If it is still the issue, try adding same Feature Service again to ArcGIS Online using Service URL with a different Name >> Share it with Public and Open data group. If it works then there is probable corruption of the item.
5. If it doesn't work create a new Open Data Group and try sharing services with that group and add that group to your site to see if issue is group specific.
This can go on, but if you can check few things for further troubleshooting;
1. How big are the datasets in question (How many features)?
2. Since when issue occurred? Did anything changed?
Hope this helps!