She had a miscarriage after one month, and that coincided with the time of her period – should she pray?

If a woman has a miscarriage before eighty-one days of pregnancy, the bleeding that happens is not regarded as nifaas; rather it is istihaadah (non-menstrual vaginal bleeding), unless it coincides with the usual time of her period, in which case it is menstrual bleeding.

Istihaadah does not prevent a woman from praying, fasting and doing tawaaf, but she has to do wudoo’ at the time of each prayer.

As this time is close to the time of your period, and the menses may come late or come early, and the basic principle is that what comes out of a woman is menses, then it seems that you should regard the bleeding you have as menstrual, and whatever lasts longer than the five days of your normal period should also be regarded as menses, so long as it does not go beyond the longest time for menses, which is fifteen days according to the majority, or last longer than a month according to those scholars who do not think that there is any maximum length of menses, such as Shaykh al-Islam Ibn Taymiyah and those who agreed with him.

See also the answer to question no. 65903.

To sum up, the bleeding that you are experiencing is menses, and is not to be regarded as istihaadah unless the bleeding lasts for more than fifteen days, at which point you should do ghusl and pray.

And Allah knows best.