Is it permissible for a man to speak with a non-mahram woman about an important matter more than what is necessary, such as speaking with her every day more than once?

It is permissible to speak with a non-mahram (definition) woman when there is a recognized need, and to the extent of that need, as long as there is assurance against fitna (temptation/strife), even if the conversation is daily. However, exceeding the extent of the need is a means to the forbidden and to fitna. Some scholars have explicitly prohibited speaking with a young woman without need, because it is a place where fitna is likely to occur.