Hi..
Welcome to HEALTHCARE MAGIC..
I have gone through your query and can understand your concerns..
As per your complain, it seems that you have a 
sinus infection secondary to cold leading to congestion and making you to blow your nose often..
As there must be inflammation in the 
nostril mucosa and the sinus linings due to infection,blowing of nose cause bleeding as the inflamed tissues have abundance of premature blood vessels that can get rupture and bleed on slight stimulation..
The  fleshy substance that you have blown out can be a thick mucous plug mixed with blood or can also be a small polyp that might have dislodged from the nasal or sinus lining..
I would suggest you to consult an Otolaryngologist and get evaluated and he can advise investigations like PNS VIEW X ray of the 
sinuses or MRI scan to evaluate the sinuses..
You can be advised a course of antibiotics like 
Augmentin, anti inflammatory painkillers like 
Motrin or Advil and along with you can also be advised to take decongestants like Pseudoephadrine and nasal decongestant sprays like 
Xylometazoline or 
Oxymetazoline..
You can also be advised saline nasal irrigations, steam inhalations and use a humidifier in your room.
Avoid spicy and chilled foods..
Hope this information helps..
Thanks and regards.
Dr.Honey Nandwani Arora.