Rabbis such as Abudraham have explained a woman's exemption from positive time-bound commandments (mitzvot shehazman grama) in terms of a woman's ongoing and preliminary role in supporting her children/husband in the home (see here).
I'm wondering about single-parent fathers, in which for whatever reason, the father is alone in raising his children. Certainly, the constant mitzvah of infant-care, coupled with the burden of providing an income, must be overwhelming. I suspect this is becoming more prevalent in modern times, and I am familiar with a single-parent father who works from home in order to take care of his little children.
In such a scenario, would there be any leniency to exempt the father from certain time-bound mitzvot that might be near impossible to fulfill (i.e., praying shacharit while needing to supervise an infant)? Until such a father finds a new wife or some kind of other support system, could he be exempt like a woman in certain mitzvot?
Also, has this ever come up in rabbinic literature?