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Thank you for your question and sorry for the delay, my name is Steve and I am a fully qualified vehicle technician with over thirty years in the trade, hopefully I can help you.
To tell you exactly where the starter motor is I would need the engine size, if you have the registration number this should help me to get the details I would need.
But as you have described the problem I do not think the fault is with the starter motor, this would not cause the vehicle to simply cut out and not start again.
Your problem sounds to me as though the battery has gone flat. this may be either because the battery has failed and will need replacing or it may be due to a failed alternator.
I would also check the connections to the battery and the large fuses on top of the battery which control the charging and other systems.
I hope this answers your question but do please feel free to ask any further details by replying to this thread.If this does answer your question please click accept and rate me, this allows me to continue. Even After clicking accept you can continue to ask further questions on the matter by replying to this thread.Many ThanksSteve
does the single click you get come from the engine bay or inside the vehicle?
Your VW does have an entry / start control module located to the left of the passenger footwell.
If the single click is from the engine bay it may well be from the starter. this is located under the exhaust manifold / down pipe. and you would need to check if this is getting current to the activation terminal on the solenoid. Unfortunately the starter is not easily accessible and the book time to replace it is 2.8 hours
It would also be worth checking that the engine earth cable is secure and has continuity from the engine to the body.
The cost would depend on where you are located and if you were to use an independent garage or a main dealer. The independent garage hourly rate would be around £55 - £75 per hour and the main delaers would probably be from £85 - £115 per hour.
The earth cable runs from the chassis to the engine and normally is easier to see from underneath. If you have jump cables you can eliminate this by placing one side of the jump lead on the engine and the other to a point on the body, near the top strut. This will then ensure that the engine has a good earth and if this is the problem will allow the engine to start.
But from your description it does sound like the starter motor.