I cough up a yellow mucus and it disturb me for freely breathing but once I cough it out I breath freely and also in my throat I cough out white sputum...which medicine can help me

Unable to diagnose over the internet without photos or tests, so it is necessary to pay a veterinarian:

http://articles.extension.org/pages/35195/one-of-my-cows-coughs-protrudes-her-tongue-and-breathes-with-her-mouth-open-is-this-an-indication-of

Quoting from article:

"
One of my cows coughs, protrudes her tongue, and breathes with her mouth open. Is this an indication of health issues? Beef Cattle March 06, 2008
The cow most likely has a lung disease in which inflammation elicits an irritated cough and reduced air space which encourages open-mouth breathing. Since a number of infectious and noninfectious causes are possible, professional assistance is necessary to make a specific diagnosis by physical and laboratory examinations. Infectious Lung Disease (Pneumonia) Pneumonia is a common infectious disease that is a possibility. Fever, coughing, and labored breathing are present due to inflammation and swelling of the lungs and accumulation of mucus, blood, and pus in the air passages that interfere with airflow. In an attempt to get more air, the head and neck are outstretched, and the tongue protrudes. Pneumonia is a highly complex, contagious disease and may be caused by one of several viruses in concert with various bacteria. Bacteria generally cause serious pneumonia, either as primary or as secondary invaders. Since predisposing stress factors contribute to the appearance of the disease by lowering the animal's resistance, management directed toward adverse conditions of cattle is needed to prevent more cases of pneumonia in a cowherd. The sick cow must be isolated for treatment. It is concluded the infectious agents are harbored in the cowherd; therefore, it is advisable to know by laboratory tests the specific viruses or bacteria involved to develop an effective vaccination plan. The plan should include vaccinating the cows, nursing calves, bulls, and replacements with the proper vaccines. Fog Fever (Pulmonary Emphysema and Edema) is a common noninfectious condition that is another possibility. With fog fever, fever is not present, the coughing is minimal, the respiratory distress is extensive, and the onset of symptoms is sudden. Difficult breathing is obvious with mouth breathing, extended tongue, and drooling saliva. The lung damage is the result of a toxic reaction in the lungs following an intake of a large quantity of an amino acid in lush, green grass in spring or fall. Diagnosis is based on a history of moving the cows from a dry, brown pasture to a lush, green pasture within the last 10 days. The affected cow should be appropriately treated and handled with caution to prevent ***** by suffocation brought about by exercise. The cowherd should be moved from the lush pasture and gradually returned over three weeks by feeding hay and limiting grazing time. "
One of my cows coughs protrudes her tongue and breathes with her mouth...

This is just a suggestion - you should ask your doctor about it though. One of the symptoms of gastric reflux is a very persistent dry cough (no phlegm, just a tickle). It may take a gastroscope to find the typical redness and erosion.
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A cause can be smoking , allergy, (milk) cystic fibrosis, neurosis and muscle tension,Wear of the cervical vertebrae,zenker diverticulum,angina pectoris.

7 Best Acupressure Points for Cough Relief Self Care Techniques

the doctor would not prescribe ANY antibiotics for a cough,anti biotics are for infections,Viral Bacterial Fungal Rickettsial,unless you know which one you have,do not try to self diagnose,you could be allergic and get worse,if it is for an animal,antibiotics are different for certain animals and ppl cant take animal medicine or reverse,children should be taken to a doctor,here is how to treat a cough for humans,
Cough Treatment Coughs can be treated in a variety of ways, depending on the cause of the cough. For most healthy adults, most treatments will involve self-care.

• Self-Treatment
  • keep hydrated by drinking plenty of water
  • elevate your head with extra pillows when sleeping
  • use cough drops to soothe the throat
  • gargle hot saltwater regularly to remove mucus and soothe the throat
  • avoid irritations, including smoke and dust
  • add honey or ginger to hot tea to relieve the cough and clear the airway
  • use decongestant sprays to unblock the nose and ease breathing

If there is no improvement after a long period of time, consult a doctor for examination, sometimes slightly higher than expected background and is not always solve through alternative means.
Wear good shoes
Always wear shoes and socks that breathe properly it can prevent a lot of throat complaints, especially in winter and get enough sleep, smoke as little as possible, take some more onions with meals, remove the sugar from the coffee take stevia or coconut sugar. eat less white flour products, and moderate meat and milk.

yes they will sooth your throat and open your breathing. I have found that musinex works great

Colds are basically viral infections that affect the nose, throat and sinuses. Younger kids and babies can catch up to 10 colds a year on average. Mostly colds are not dangerous and get better by themselves but if you see any of the following, you should consult with your doctor;

- A fever over 38 degrees

- Cold symptoms for more than 10 days especially with green or yellow mucus.

- Any breathing difficulties.

- Throat pain.

Fill your bowl or a sink with steaming water, add a couple of drops of eucalyptus oil and put your head over the bowl with a towel over your head to keep the steam in and breathe deeply.

Make sure you drink a lot as well and try using saline sprays to thin the mucus in your nose and to add moisture to dry membranes.