I am struggling how to define a log-likelihood based on an observed value of a quantity (data), the measured value based on a given model (model) and instead of using the errors ($\sigma_i$) in the measurement of the observed quantity, I would like weighing the likelihood with a value which is a combination of errors and another measured parameter ($w_i=\frac{1}{\sigma_i^2+\delta^2}$) where $\delta$ is a given constant value. As it can be inferred $w_i$ has reverse property, meaning it is higher for values with smaller errors mostly.

How could I substitute this value for each measurement instead of sigma in the likelihood function ?