This bleeding is not menstruation, so it does not affect the fast, so do not stop praying because of it. But you have to do wudoo’ for each prayer. 

Shaykh Ibn ‘Uthaymeen was asked: if a small spot of blood comes out of a woman during the day in Ramadaan, and this (spotting) continues throughout the month of Ramadaan and she fasts, is her fast valid? 

He replied: 

Yes, her fast is valid, and this spotting does not matter because it is from a vein. It was narrated that ‘Ali ibn Abi Taalib (may Allaah be pleased with him) said: “These spots that are like a nosebleed are not menstruation.”  

Sittoon Su’aalan ‘an Ahkaam al-Hayd.

The Standing Committee was asked: Sometimes my period lasts nine days and sometimes ten. When I become pure (i.e., my period ends) and I get up to do housework, it comes back, starting and stopping. If it comes back after the time stated in sharee’ah that a period may last, is it permissible for me to fast, pray and do ‘umrah? 

They replied: 

The length of the period in your case is the length of time your period usually lasts, which is ten or nine days. When the bleeding stops after nine or ten days, then do ghusl and fast and pray, and do tawaaf for Hajj or ‘umrah, or a voluntary fast, and it is permissible for your husband to have intercourse with you. Any bleeding that resumes after the period is over because of activities that you do or any other reason is not menstrual blood, rather it is due to some sickness or problem, so it does not mean that you cannot pray, fast, do tawaaf or any other acts of worship. Rather you should wash it away, like any other impurity, then do wudoo’ for every prayer, and pray and do tawaaf around the Ka’bah, and read Qur’aan. 

Fataawa al-Lajnah al-Daa’imah, 5/426.