There are rumors that there was a secret love affair between Rasputin and the Tsarina during his time at court (how truthful those rumors are is another thing)
Regardless, in a situation where there was at least a sexual relationship between the two, what might the political effects be if said relationship produced a healthy son?
Would the Romanovs pass him off as
Nicholas' to save face?
Would Rasputin have a door into the aristocracy?
What of the Tsarina if Nicholas discovers it?