I have simple Sitecore8/PowerShell script where I delete image child item and publish parent item:
$child1 = Get-Item $child1Path
Remove-Item $child1 -Force -Permanently
Publish-Item -Item $parent1 -Recurse -PublishMode SingleItem -Language "en*"
No deletion and no errors while running.
How can I find out why it's not deleting ?
You can use the Remove-Item with the -Path parameter. Or use
$child1 | Remove-Item
Apparently you script don't throw an error but the remove is not working that way. See the documentation https://doc.sitecorepowershell.com/working-with-items