I thought I backed up my iphone 4s, but I only synced it. Can I restore somehow?

I meant to do the backup before jailbreaking. I did an untethered jb with evasion. Now the jailbreak has gone screwy- the new icons have all disappeared from my home screen, including cydia. I've tried to put the cydia back on, but it doesn't work. At this point, I feel the only way to fix my issues is to restore to previous state, but I have no option to restore a backup. I do have info in my itunes backup folder, Is there any way to restore my phone even partially with this data? I attached a jpeg with an img of the files I have, I've copied these from itunes to another folder for safekeeping

Firstly- never move system files from their original location as for some reason they can become un-usable if you ever need to use it again

try soft reset on the iPhone to factory setting by holding the power and home button till the screen goes black and the apple logo appears(this worked when my iPod went into boot loop and wouldn't restore via iTunes)

Also check if you have iCloud backup? The backup does work quite well but takes time depending upon your net speed.

Now in iTunes since you don't have backup you will to restore it to factory settings which requires iTunes to download the iPhone OS and reinstall it. Mind you that all data will be wiped from the phone.

Before restoring your iPhone to factory setting give this method a try:-
Try backing up your iPhone to iTunes in its current condition if you have data you absolutely need, it may or may not back up Jail-broken iPhone and also if it does back up your iPhone it may or may not restore data on a non-jalbroken iPhone. If the back up works and doesn't restore on a non-jailbroken iPhone, jailbreak it then try restoring the back up.
Also try the above method with iCoud back up instead of iTunes backup.