Since the line with equation $y=mx-50$ passes through the point $(a, 0)$, then $0=ma-50$ or $ma=50$. Since $m$ and $a$ are positive integers whose product is 50, then $m$ and $a$ are divisor pair of 50. Therefore, the possible values of $m$ are the positive divisors of 50, which are $1,2,5,10,25,50$. The sum of the possible values of $m$ is thus $1+2+5+10+25+50=93$.