Offset vs fixed factor in a mixed poisson model

Thank you for the explanation. Si I should estimate the parameter for my covariable. Is it correct to run the model with only the log-transformed parameter as an 
explanatory variable and look at the estimate? I come to an estimated parameter of 0.8, which is quite close to 1. Is there any threshold, under which it is not 
recommanded to use the variable as an offset? How can I control for my covariable if not as an offset since mixed-models are not performing sequential tests 
(right?), such as in an anova?
Best,
Valerie
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