Is it permissible for a woman to continue working long hours in a mixed-gender environment due to financial need and her husband's desire, while trying to avoid mixing as much as possible?

It is impermissible for a woman to work in a place where she mixes with men, due to the Sharia-based prohibitions therein and the resulting exposure to corruption. An exception is made for necessity that cannot be averted except by working in such a place. What the questioner mentioned is not considered a necessity that permits committing what is forbidden.