The code below gives the following results for the fit.

The result on the parameter I am interested in (p0) seems to depend on the scaling of the histogram. The uncertainty is especially affected. Is there any way to eliminate this dependence? It seems the fit is not using the sum of weights information?

FCN=479.114 FROM MIGRAD STATUS=CONVERGED 127 CALLS 128 TOTAL
EDM=5.83227e-06 STRATEGY= 1 ERROR MATRIX ACCURATE
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 1.50677e+00 7.95085e-03 8.51393e-05 6.03274e-02
2 p1 7.50069e+02 3.35441e+00 3.58871e-02 -1.00807e-03
FCN=40.3398 FROM MIGRAD STATUS=CONVERGED 38 CALLS 39 TOTAL
EDM=0 STRATEGY= 1 ERROR MATRIX UNCERTAINTY 100.0 per cent
EXT PARAMETER APPROXIMATE STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 p0 1.47044e+00 9.50993e-05 2.73655e+00 3.02586e+08
2 p1 -3.70608e+07 3.18227e+02 -3.70616e+07 0.00000e+00

Minuit fails in the scaled fit (the value of the second parameter is totally wrong). It needs to be investigated.
I have tried with Minuit2 and it works fine. For using Minuit2 just add the following line before fitting:

Somehow I hope that the uncertainty on the second fit parameter 0 should be the same as the uncertainty on the first fit parameter 0? i.e. 1.502156 +/- 0.0079677602 ?

This is expected, because, when doing a Poisson likelihood fit you should not scale the histograms. A scaling factor will not influence the resulting value, but it will change the resulting error, since the rule DeltaL=1 (or 0.5) is not valid anymore. You should then scale also the DeltaL value.

If you use a simple least-square fit, you will get the same error in both cases.

Is it possible then to compute an accurate uncertainty for weighted events? The problem is as follows: We have MC simulations for different mass regions, with different luminosities. They are weighted to give a continuous mass spectrum. The graph we are fitting to is a distribution of theta in a given mass bin. Then we take a bin in (smeared) mass, so we have events with different weights ending up in this bin. Is there a sensible way to compute the uncertainty on this parameter (0)?

[edit] sorry, spoke in jargon. Edited for clarity.

You can estimate the error in the single bin, by using the sum of the square of all the weights ending-up in the bin and then doing a least square fit.
If you want to do a likelihood fit, I think there exist some way for handling the weights. Something is described in the Statistics books from F. James