If you see a red exclamation point icon in your Perusall gradebook, that means that grade will not sync back automatically to the LMS. In order for the grade sync to work, students have to launch each Perusall assignment from an assignment-specific link in your LMS (i.e., one with a name that matches the title of an assignment in Perusall). During that launch process, the LMS passes information to Perusall that tells Perusall where to send the grade for that student back to. If you create an assignment-specific link, you can test that it is set up correctly by clicking it in your LMS -- ensure that it launches directly into the corresponding assignment in Perusall (instead of going to the Perusall home page).
If you have a grade for a student that isn't syncing back to Perusall, it may be because the student launched Perusall from a generic LTI link and navigated to an assignment within Perusall (so Perusall won't be able to sync the student's grade on that assignment back to Perusall).
However, this can be fixed retroactively -- if the student launches the Perusall assignment from the corresponding assignment-specific link in the LMS (and then just closes the browser, if the assignment is already complete), that will allow the LMS to send the information to Perusall it needs to do a grade sync, and that student's grade will then be sent back to the LMS. (Most grades sync immediately, although it may take up to an hour for a grade update to be reflected in the LMS.)
If you see an orange exclamation point icon in your Perusall gradebook, that means that Perusall encountered an error when trying to communicate with your institution's LMS. Hover your mouse over the orange exclamation point icon to get more information about the error. Usually, your IT department can correct the error. Once the error has been corrected, click the "click here to try to resync" link to try again; if successful, the scores should show up in your LMS gradebook within a couple of hours.
If you don't see any exclamation point icon next to the non-syncing score in your Perusall gradebook, then you may just need to release the grade so students can see it; scores will not be sent to the LMS until you release them to students. Click the Release Grades button at the bottom of the corresponding gradebook column to release. You may also want to set up an automatic grade release policy under Settings > General if you do not want to release scores manually.