Cough. For a 10 month old, there are no medications for cough or cold symptoms. If this cough is new, and due to a cold, you can use saline nasal spray, and a nasal aspirator to help with congestion. If the cough is more chronic, your child should be evaluated.
No meds. The fda has restricted cough meds in children under 4. After age 1 you can give honey based cough meds. Under one year use a vaporizer. Use nasal saline plus bulb syringe. Cleannoz and nosefrida are new gadgets to replace the bulb. Also have her sleep with head elevated. Vicks is ok to use. If rapid or labored breathing see doc. If not drinking, lethargic, very fussy, hi fever etc. See doc.
Support. Water, water, water is my favorite treatment for most cough/colds. Add water to the air with a clean humidifier or vaporizer (no added meds). Put saline (salt water) in her nose to help clear the mucus. Use a nasal aspirator (such as nosefrida) to clear the nose. Be sure she drinks plenty of fluids. Go to doc if she sucks ribs/belly with breathing or is breathing 60+ times / min, not drinking.
10 month old & cough. You can take steps to ease the symptoms of cough. If the cough is due to a common cold, try plenty of fluids and keep the air moist using a humidifier. If the nasal passages are congested, try saline nose drops to loosen thick nasal mucus, and suction the nose. For the cough due to acid reflux, try smaller, frequent feedings, frequent burping, keeping the baby upright during/after feedings.