'Problem could be related to Injection Pump and engine timing. Since the IP is hooked to the timing chain, it would be prudent to make sure you don't have some chain stretch (wear) issues. If you have never changed the timing chain, you are most probably due. Since you're not hearing anything at idle, it will be difficult to troubleshoot. But making sure everything is as close to specification (timing) as possible will give you a good basis for other more progressive (expensive) changes, like rebuilding or changing injectors.

Good luck!

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1976 240D
1987 560SL
2007 E320 Bluetec
1998 C280 (now son’s car)
1982 240D Manual - Sold