Hi confusedpenguin,
If you used the Windows refresh option, the bookmarks might be recoverable. If you check your C:\ drive in File Explorer, do you have a folder called "Windows.old"? If so, the old bookmarks files may have been saved here when Windows refreshed.
If you have the Windows.old folder, please look for files called "Bookmarks" and/or "Bookmarks.bak" in the folder C:\Windows.old\Users\[your_username]\AppData\Local\AVAST Software\Browser\User Data\Default. If one or both of these files exist, please make sure Secure Browser is closed, and then copy and paste the file/s into the current Secure Browser folder (C:\Users\[your_username]\AppData\Local\AVAST Software\Browser\User Data\Default), and you should get access back to the bookmarks the nest time you open Secure Browser.
Please note, if you have any version of Avast Antivirus installed, you will first need to disable Avast Self-Defense, as the Secure Browser folders are protected and will not allow you to modify them. To do this, please open your Avast Antivirus application > click "Menu" in the top right corner > click "Settings" > select "Troubleshooting" in the central column > un-check "Enable Self-Defense" > click "Yes" on the warning pop-up. Please remember to re-enable this by checking this box again after pasting the bookmark files.