It might be due to nondevelopment or poor development of nasolacrimal duct. It's a congenital problem,Usually self limiting. The only support of treatment is not to let the tear or discharge get deposited there. This can be done by gentle massage around the lacrmal area and put some antibiotics eye drops 2hrly e.g. moxifloxacin,just to prevent the deposits (If any)from being infected. You can learn the lacrimal massage by consulting any ophthalmologist nearby.