This would be easier if *all* components had reference designators.
U1's 10 K output resistor, R12, R13, and R14 form a variable attenuator. The percentage of U1's output that appears at the U2 input varies with the adjustment of R12. Likewise, the offset voltage source at the R12 wiper is attenuated by over 50% at the U2 input (assuming the pot is not at its extreme rotation).
If you want to stick with this arrangement (as opposed to adding a true (inverting) summing stage before an inverting output buffer), then change the voltage reference to an adjustable type. This will present a consistant zero-ish ohm source impedance for the offset summing resistor that is independent of the wiper position. There still will be a 50% attenuation of the U1 output. National, ADI, and Linear Tech make adjustable references.
NOTE: Thinking this further through, this circuit has a problem that has no simple fix:
If the U1 output is 0 V and the offset is 4 V, what appears at the U2 input is 2 V (50% of the offset).
If the U1 output is 2 V and the offset is 4 V, what appears at the U2 input is 3 V, not 6 V.
If the U1 output is 4 V and the offset is 4 V, what appears at the U2 input is 4 V, not 8 V.
Because the U2 input is not a virtual ground, it allows interaction between the two currents; either current can go either direction through either resistor. The more narrow the U1 output voltage range, the less the interaction will be noticeable; but it never will go away and the voltage seen by U2 never will be a simple offset of the voltge at the output of U1. The error curve will be logarithmic.
ak