Offset vs fixed factor in a mixed poisson model

Dear subscribers, 
I am tested the effect of a factor on a count variable using a poisson mixed model. I know that my response variable is linearly influenced by an other variable so 
that I would like to remove the effect of this second variable to see the true effect of my factor. In an anova, it is usual to enter the covariable first in the model and 
use a sequential test (type I SS). However I am a bit confused how to control for this covariable in my mixed-poisson model. If I just give the covariable as an 
additional fixed variable, my factor is highly significant. If I put it instead as an offset, the factor is not significant at all. I think that it is better to use offset, but I must 
admit that the underlying "theory" is not clear for me. I was also wondering if we can specify multiple offsets and if there was some "rule of thumb" in the maximal 
number that can be included. Thank you very much. 
Best,
Valerie
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