Single word for the phrase "Investing more money to save the money already invested"?

I don't believe there is a single word (why should there be?), but Sunk Costs fits the bill pretty well. Note that many economists refer to the 'Sunk Costs Fallacy', and there is a proverb referring to 'throwing good money after bad'.

You may be looking for a cash infusion. This often connotes spending money to keep a company from failing.

There have been quite a few government bailouts.

The person providing the infusion may be a white knight.

If the infusion is likely to fail, then there are a host of proverbial phrases:

Throwing money down a rat hole.

A waste of perfectly good money.

Escalation of commitment (from the title of a paper).

In for a dime, in for a dollar.

In for a penny, in for a pound.

One relevant expression is "throwing good money after bad (money)". This imples that you've already lost some of the money (the bad money), and that now you're adding more money which you could still use for other things.

How about "doubling down" - strengthen one's commitment to a particular strategy or course of action, typically one that is potentially risky.