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I am using the SD Card Copier accessory to make a micro-sd card back-up. The back-up card (in a micro-sd to USB adapter) is recognized by the copier as sdb, and I know that the copying is OK, because I can replace the OS micro-sd card by the back-up card, and the Pi boots successfully with everything in place.

On my previous Pi (model 3B) I could check the back-up card by simply removing it from the USB port (as it was unmounted) and re-plugging it so that I could inspect it via pcmanfm.

On the Pi 4 the re-plugged micro-sd back-up card is not recognized by the system and is impossible to mount or read. I can read it in the File Manager on my desktop PC, so there is presumably no problem with the micro-sd card itself.

Is there anything I can do to read the back-up card on the Pi 4?

I don't think the Pi model is a factor here.

When you made your backup, did you select New partition UUIDs?

If not, that's likely the problem.
SD Card Copier help says this:

Under Raspbian Stretch and later versions, you cannot mount two partitions with the same UUID, so you will not be able to mount a cloned SD card when booted from the disk from which it was cloned. If you need to do this, check the "New Partition UUIDs" box before copying.

You can change the UUID with this command:

sudo tune2fs /dev/sdb2 -U random

After that, unplug the microSD adapter, then reinsert it. It should auto-mount from now on. If not, try rebooting.

I regularly read SD Cards on my Pi4, so this is possible.

(The USB3 ports on the Pi4 have some limitations with USB2 devices, although I would not expect this to affect a SD Card reader, but have had occasional problems and usually use one of the USB2 ports.)

I have experienced only one problem which proved to be due to a faulty Card.

I need to manually mount the Card, and can post the script I use to do this.

#!/bin/bash
# 2017-05-06
# 2018-11-18

BOOT_MOUNT='/mnt/SDA1'
ROOT_MOUNT='/mnt/SDA2'

# Check/create Mount Points
if [ ! -e $BOOT_MOUNT ]; then
 mkdir $BOOT_MOUNT
fi
if [ ! -e $ROOT_MOUNT ]; then
 mkdir $ROOT_MOUNT
fi
 echo "mounts " $BOOT_MOUNT $ROOT_MOUNT
if [ -e /dev/sda ]; then
 SD1='/dev/sda1'
 SD2='/dev/sda2'
else
 SD1='/dev/sdb1'
 SD2='/dev/sdb2'
fi
echo $SD
# Mount Partitions
if ! $(mountpoint -q $BOOT_MOUNT); then
 mount $SD1 $BOOT_MOUNT # mount partition containing boot files
fi
if ! $(mountpoint -q $ROOT_MOUNT); then
 mount $SD2 $ROOT_MOUNT # mount root partition containing OS files
fi

The following unmounts SD Cards

#!/bin/bash
# 2017-05-06
# 2018-11-18

BOOT_MOUNT='/mnt/SDA1'
ROOT_MOUNT='/mnt/SDA2'

umount $SD1 $BOOT_MOUNT # mount partition containing boot files
umount $SD2 $ROOT_MOUNT # mount root partition containing OS files

The point raised by Botspot is valid.

You should give SD Card copies a unique PARTUUID which needs to be changed in several places. (SD Card Copier has an option to do this.)

The following script set-diskid will set PARTUUID on a booted system in all the correct places.

It can be invoked with sudo set-diskid -n

NOTE you need to reboot after running the script.

#!/bin/bash

errexit()

 echo ""
 echo "$1"
 echo ""
 exit 1


usage()
 diskid]"


if [ $(id -u) -ne 0 ]; then
 errexit "$0 must be run as root user"
fi

PTUUID="$1"
if [ "$PTUUID" = "" ]; then
 usage
fi
if [ "$PTUUID" = "-n" ]; then
 echo $PTUUID 
 PTUUID=$(uuid | cut -c-8)
fi
PTUUID="$(tr [A-Z] [a-z] <<< "$PTUUID")"
if [[ ! "$PTUUID" =~ ^[[:xdigit:]]8$ ]]; then
 errexit "Invalid DiskID: $PTUUID"
fi
echo ""
echo -n "Set DiskID to $PTUUID on /dev/mmcblk0 (y/n)? "
while read -r -n 1 -s answer; do
 if [[ "$answer" = [yYnN] ]]; then
 echo "$answer"
 if [[ "$answer" = [yY] ]]; then
 break
 else
 errexit "Aborted"
 fi
 fi
done
echo ""
fdisk /dev/mmcblk0 <<EOF > /dev/null
p
x
i
0x$PTUUID
r
p
w
EOF
sync
PARTUUID="$(sed -n 's|^.*PARTUUID=(S+)s.*|1|p' /boot/cmdline.txt)"
if [ "$PARTUUID" != "" ]; then
 sed -i "s|PARTUUID=S+s|PARTUUID=$PTUUID-02 |" /boot/cmdline.txt
 sed -i "s|$PARTUUID:0:($#PARTUUID - 1)|$PTUUID-0|" /etc/fstab
fi
sync

NOTE PARTUUID and UUID are different entities.

UUID is a 128-bit number used to identify information in computer systems, in particular it is used to identify partitions on GPT & Linux filesystem.

MBR does not support partition UUIDs, but Linux supports PARTUUID for MBR partitions.

The format is SSSSSSSS-PP, where SSSSSSSS is a 32-bit MBR disk signature (stored in the MBR label-id field), and PP is a partition number.