based on that ddrescue was not successful, the image file is incomplete.

This is very likely, yes. Reported image size 4 GB is rather small for home partition.

e2fsck is trying to read what would be the last block in a ~25.9GB (24.12 GiB) filesystem. That image file is too small to hold a filesystem of that size.

from the bad hard drive
sudo ./ddrescue -n /dev/sda5 /media/HP/my_file2.img

that's how I created it.

yeah, because my usb stick is not big enough.
Can that 4GB image file be mounted?

You cannot perform any work on partial image, you need the image of full partition.

OK, I use ddrescue and saved to a backup USB drive which is big enough.
Now, I use a healthy system and fdisk my_file2.img

fdisk -l my_file2.img

Disk my_file2.img: 58.7 GB, 58730741760 bytes
255 heads, 63 sectors/track, 7140 cylinders, total 114708480 sectors
Units = sectors of 1 * 512 = 512 bytes
Sector size (logical/physical): 512 bytes / 512 bytes
I/O size (minimum/optimal): 512 bytes / 512 bytes
Disk identifier: 0x00000000

Disk my_file2.img doesn't contain a valid partition table
==========================

now , how do I mount the rescued image? It prompts for file type.
I did use -t auto but it didn't help.

I did file my_file2.img
my_file2.img: data

How big was the original disk? 58.7 GB is not a size of any disk drive manufactured in the last 10 years. The size provided would be between 60 and 100GB and seems small.

I think this is an image of home partition.

what will fdisk -l /dev/sda respond? (what was the original ?)

You copied just one partition. There is no partition table. That is normal.

Now you need to run fsck on the image file:If it looks like it is behaving reasonably and you're geting tired of typing "y" to all the prompts, interrupt it with ctrl-c, then run it again with the "-y" option.Once e2fsck has recovered the primary super block and patched up the rest of the filesystem, you should be able to mount the image with the "-o loop" option.

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