Check the settings in Tools>Options>User interface>Reporting/logging.  There is a "Hide the Pegasus Mail taskbar entry when minimized" setting in there that might have something to do with what you are seeing.  Clicking the Desktop icon in the Quicklaunch area minimizes all windows and shows the Desktop (same as the key combination Window Key + D) so having this setting enabled would hide the Pegasus Mail taskbar entry when Pegasus Mail gets minimized.

I am clueless the change you see in Task Manager and can't duplicate it here (XP SP3).  The MAILBOXP.LCK file indicates that the mailbox is in use by an instance of Pegasus Mail but I don't know why you are seeing it listed in the Task Manager Processes tab.

When I have Pmail running normally there is, correctly, an entry for it in the
taskbar and the Windows tray. There are correct entries in Windows Task
Manager/Applications and Processes

If I strike the Desktop icon in the Quicklaunch area it correctly switches to
the Desktop and leaves the Pmail icon in the tray however it removes the Pegasus
Mail entry from the Taskbar and the entry for Pegasus Mail in Windows Task
Manager/Applications no longer appears but the entry in Windows Task
Manager/Processes remains as well as MAILBOXP.LCK.

I suspect that this is something to do with an entry in PMAIL.INI because, if
I use an older version PMAIL.INI from a backup, it behaves properly when
striking the Desktop icon. Unfortunately the older version has failings in
other respects so a lot of dicking around would be required to get it working
as required.

Any ideas of what entry in PMAIL.INI might be causing the problem?

Windows XP SP3, Pmail v4.63