Assuming it is one of these ..
https://www.machinemart.co.uk/p/clarke- ... ing-lathe/
and as you say it does not have a Morse Taper fitting, you are somewhat limited in ways of holding the workpiece.
I believe that apart from the fixed drive centre and faceplate you cannot fix any other accessories i.e. mandrels and chucks*. So if you are not turning spindles your only other method of holding the wood is using the faceplate.
In order to turn pens I think your simplest (and only?) solution would be fix a block of wood to you face plate, maybe a 3" cube, and drill a hole* in the centre to accept the Morse Taper. The size of the hole should be the diameter of the MT at maybe 5mm from the fat end. This should allow you to hold the MT securely whilst the tailstock holds the other end of the mandrel parallel and provides the 'push' to hold the mandrel in place.
*Unfortunately another shortcoming of your lathe is that you cannot fit a Jacobs (drill) Chuck in the tailstock, which makes drilling centrally into the workpiece a simple operation.
'Drill' your hole, or at least the beginning of it using a gouge (see at about 2minutes into this video
https://www.youtube.com/watch?v=YPM2vQtuBBY )
You could purchase any pen mandrel of your choice, (Axminster, turners retreat, rutlands etc).