Mark, don't despair! In fact, congratulations, that broken clip you have is the probably the most common 944 ailment so it was going to happen sometime. Chances are the clip has been broken for a long time.
With the clip broken your heater will 'fail' to full heat at all times so no matter where you turn the temp knob on the heater control panel you will only get max. hot air from the system - we all assume this is to ensure that if the heater fails in the middle of a German winter you don't freeze to death!
The plastic clip is (probably still) only a £1 or so from an OPC.
Keep the metal clip as you'll need to reuse it. It's a simple DIY refit as per Clarks.
As Endoman says you need to follow the rubber tube (about 12 inches long) that leads from behind, and next to, the cig lighter to the small interior temperature sensor unit which contains an internal fan. From your photo I can see the rubber tube - it has the 944 part of the part number visible in your photo - so follow this tube until you get to the other end and it should be firmly connected to the interior temp sensor unit which looks like this>
https://www.ebay.co.uk/itm/Porsche-944-1986-1991-Glove-Box-Air-Fan-Temperature-Sensor-Bosch-0130002809/331651237982?fits=Car+Make%3APorsche%7CModel%3A944&epid=1709548070&hash=item4d37f4485e:g:9PkAAOSwQTVV8Y0wFrom the pics in the above eBay link you'll see the fan is sort of an internal unit with plastic fins - it's just above the silver Bosch sticker.
Like you, my 1989 S2 had a noisy interior temp fan and the broken clip. To remove/improve the noise I squirted a load of WD-40 (using one of the cans with the straw attachment) into the general area of the fan and its spindle. The unit is a pig to get to or remove so I just squirted away while leaving it in situ. I also discovered I had no rubber tube (not refitted after a dash change years before by previous owner), so this made the noise worse as the fan was working more frequently and the tube helps mask the fans noise. So, make sure your rubber tube is firmly connected at both ends.
After connecting the rubber tube correctly, fitting a new heater clip and after the liberal spray of WD-40, my interior temp fan motor is
much quieter although its noise is still clearly audible for about 30-60 seconds on first start up as the temp sensor does its thing, but it's only a faint buzzing which goes away once the fan stops. I notice that in winter the fan works (and therefore buzzes faintly) for slightly longer than in summer. I also never have the temp knob much above the 21oC mark in winter so this might help as I'm not asking the heating system to do much.
My unit does not come on and off during a journey - if it does, I don't hear it above the general road and engine noise.
In short - first fix your clip and ensure the rubber tube is secure at both ends then spray some WD-40 in the fan area and see if that quietens things down. However, even then, expect some slight noticeable noise on start up for a minute or so. Don't refit your glove box inner tray until you've road tested and be careful with the glove box lid arms on refitting as the plastic will be brittle in the cold weather.
Hope this helps!
Mark