Voltage drop across a poor connection
If...

If the meter is 10Mohm then this implies 160kohm through wire??

If the complete situation is as you described, then that is exactly what it means.
However...
The input impedance of the multimeter is not constant with respect to input voltage and is not exactly 10.000000 Megaohms. Your resistance assessment is sensitive to both voltage measurement accuracy and the (unknown) input impedance of the multimeter.
That's why, wire characterization is done with a significant current and a precision (e.g. known) load.
Why not...
If you have a multimeter, why not just measure the resistance of the wire directly? Disconnect it from the source and use the meter's resistance measurement setting?