Is it permissible for me to pray Taraweeh when I am pure, but I have a brownish discharge?.
If this brownish discharge comes before you are purified from menses, then it is part of the menses, so it is not permissible to pray or fast when you have this.
If it comes after your menses has ended then it is not menses, and it does not mean that you cannot pray or fast.
It says in Fataawa al-Shaykh Ibn Baaz (10/208):
If this brownish or yellowish discharge comes after becoming pure from menses, then it is not regarded as menses, rather the ruling on it is the same as the ruling on istihaadah (non-menstrual vaginal bleeding). You have to wash it off yourself every time, and do wudoo’ and pray and fast, and do not regard it as menses. You are also permissible for your husband, because Umm ‘Atiyyah (may Allaah be pleased with her) said: “We did not regard brownish or yellowish discharge after the period ended as being anything that mattered.” Narrated by al-Bukhaari in his Saheeh (320). End quote.
Based on this, there is nothing wrong with your praying Taraweeh in the mosque even though you have this brownish or yellowish discharge, but that is subject to the condition that you are certain that your period has ended.
And Allaah knows best.