Testing for manganese in water
Best way is sending a sample for Atomic Adsorption Spectroscopy (AAS). You can get the concentration for $\ce{Mn, Pb, Fe, Cr, Ni}$, and other water goodies. Contact a local university or lab and get a price quote for a test. 
Or get a flame spectroscope and DIY https://web.pdx.edu/~atkinsdb/teach/427/Expt-AtomicSpec.pdf