After Windows 10 version 1809 update Build 17763.1, my Backup and Recovery now shows error 0x80070013 and will not backup as before

Because Backup & Restore continues to work fine for me in 1809, there's got to be a difference between your setup & mine. But so far I haven't detected it. Also, what could be different about how it ran for you in 1803 & how it runs now? We have to investigate what it takes to make a shadow copy especially. Maybe something regarding that didn't go well in your upgrade.

Yes it seems to run about 98% and then stops. I will look in to the steps you outlined but I hope MS can resolve this issue. If the W&RB is going to be included in the 1809 release then it should function as before. I know I am not the only one with the issue.

Still, I'm leaning toward it being a problem with the making of that shadow copy (which it says is on the destination/external drive). It gets that far for you before crashing?

(1) I meant to ask the size of your Windows partition (C:) not the folder. But it's probably OK.

(2) Too bad that "error" log is empty for you too.

(3) I do not see that you have a Recovery Tools partition. (At least, Backup & Restore did not make an image of it.) Have you got a file named WinRE.wim, likely in a folder named C:\Windows\System32\Recovery? If so, that is your Recovery Environment which will load when you Shift+Restart. Otherwise, you don't have one. Better make a Recovery USB (ask Cortana for Recovery Drive).

(4) Your backup log indicates your EFI partition (if yours is a UEFI install) & your Windows partition were successfully backed up. Do you know how to take System Image Recovery to the brink just to see whether it is offered for restore?

https://www.winhelp.us/restore-a-system-image-in-windows-7.html

Restore a System Image Backup in Windows 7, 8, 8.1 and 10

Note: That is a 3rd-party site. Looks knowledgeable, but I know nothing (good or bad) of any downloads or links that are there.

(a) Plug in the external HDD that holds the image.

(b) Shift+Restart for the Choose an Option screen. Or boot to the Recovery drive through BIOS.

(c) Click "Troubleshoot > Advanced Options > System Image Recovery".

(d) Keep clicking until you see it is offered - then back out.

VincenteMartinezU did say he successfully restored his on page 2. (but I wouldn't do it just for fun.)

(5) Here is how to mount the image of C:.

(a) Open the WindowsImageBackup folder on the external HDD that holds it.

(b) Get to the Backup folder with the proper date.

(c) Double-click the largest .vhdx in that folder. That will likely be the C: partition.

It might mount & open in a new File Explorer window at that point. But...

(d) If you get an error message, click "OK"...

(e) Right-click START, & select Disk Management.

     You will see it has mounted, but without a letter...

(f) Right-click the one that has no letter, & select "Change drive letter & Paths", then click "Add". Either "OK" the one offered, or select your own. Disk Management will now show a letter...

(g) You may now open the mounted image in File Explorer. Look good?

(h) When done, right-click the drive in File Explorer, & select "Eject".

I find that, after going through the error once, the next time the .vhdx will just mount & open.

(6) I think it is a problem with the making of that shadow copy that is thwarting Backup & Restore for you. But your Services settings seem OK. More study is required.

The backup is moving 29GB and shadow copy at 15% would try to write to Desktop and would get the 0X80070013.  It takes a little over an hour to complete.  The WIFI on the Laptop is running around 100mps and Desktop is Ethernet at 1GB.  The data on the Laptop is a C drive only and the Desktop has multi drives.  The data is being writen on a non system drive.

The Volume Shadow Copy and Remote Procedure Call is that same as your views.

Hope this helps!