I read somewhere that this was the case. I get what profit margins and cost-push inflation are, but don't see how increase in marginal profit causes cost-push inflation? Also, how does increase in money supply cause cost-push inflation?
You can think on prices like $p = (1+m)w$, which means that prices are the value of inputs ($w$) multiplied by the margin ($1+m$) of the firms. So it is clear that raising firms margin would increase prices. This is cost-push inflation because the margin are part of the firm´s costs, it is a payment to a factor, like capital or entrepreneurship.
A raise in money supply don´t cause cost-push inflation directly, but causes demand-push inflation, which can increase also increase the prices of productive factors and inputs to production (w) as well, if firms can pass this increases in their cost to produce to their price, then you have a channel of cost-push inflation.
Because of the higher costs firm must raise their prices, which definitely leads to inflation. As well as, higher marginal costs lead to higher marginal profits.
What kind of events cause higher marginal costs?
Demand shock: there is a huge demand for products in the market, so the firm decides to produce more exceeding his optimal cost structure. He must increase his prices, too.
Supply shock: Accidentally, the prices of capital or labour become expensive. At the end, he must increase the price of products again.
This phenomenon leads to overheating spiral: higher profits cause higher costs, and higher costs cause higher profits. The government must set higher treasury rates to kill off the inflation monster.
Sub-question. Increasing the money supply leads to the reverse effect of the former. Higher money supply increases the nominal rates, which give birth to crowding-out effect, so the aggregate output diminishes. As a result of lack of demand, the prices of production factor sink, which decrease the level of costs.