A pedantic point to start with:
The Royal Mint (in Llantrisant, Wales) only produces coins. Banknotes are printed by the Bank of England (or, more accurately, on behalf of the Bank, by De La Rue Currency in Loughton, Essex).
Printing extra money to try to solve economic problems is known as 'quantitative easing'. In modern times, where moving money around the world is done largely by simply altering the balances on computers (rather than physically moving the cash), it's no longer necessary to actually print extra banknotes; governments simply increase the balances shown in their books.
Many governments (including the present UK government) have tried quantitative easing to solve their problems. (Over the past few years, the Bank of England has 'created' billions of pounds of 'extra' money, doing exactly what you've suggested).
The problem is that money has to be backed up by gold (and other) reserves. Printing extra money (or juggling the figures on computers) doesn't change the value of those reserves; it just makes each pound worth less on international markets. That leads to rising petrol prices for our cars because we need to use more of our (lower value) pounds to pay for the fuel. It also increases the prices of all of our other imports, such as wheat (so bread prices go up).
If everything costs more, people can't buy as much, so our manufacturers can't sell as much. That can result in job losses for the employees of those manufacturers. So it's easy to make matters WORSE through quantitative easing if it's done to any great extent. (The Government's current policy is that it should only be a SHORT-TERM measure to boost the economy, with the extra money gradually being withdrawn from circulation later on to avoid the sort of problems I've mentioned).
If things really go wrong, MASSIVE inflation can occur. For example, we might expect prices to rise by anywhere between (say) 10% and (perhaps) 50% over a 9 year period. But in 1923 prices in Germany (where the government tried to pay off war debts by printing money) where 726 BILLION times those of 1914. (On a single day the price of a loaf of bread at teatime could be 20 or 30 times what it was at breakfast-time; people were having to fill wheelbarrows with banknotes just to buy a single loaf).
Chris