if its a distributor pump it can be worn pump seals, sticky anti-shudder valve or fuel cut valve, or a leak somewhere on the fuel system, allowing air to enter overnight and unbleed a line, or if its a common rail system the rail may not be holding pressure overnight, try priming with a couple of key turns before starting and see if that helps.
There should be a valve on a crtdi to hold fuel in the rail indefinately whilst off, check your manuals for the one one your motor, my only audi knowledge is on the AFN / AHU diesels (like mine lol) 1.9 TDi 110 etc, but hope that helps.
Obviously heater plugs have a great significance to starting but only in cold weather, they only blink when the temp is above 9oC and above 12 or something they dont even come on with a Di, but common rail motors use them during and after starting to improve cold combustion (smoke control), Di's dont need this as the injection pressures are much lower.