"Can government have an influence on a country's exchange rate?"

That's a good question and one of great importance to currency traders. The answer is an emphatic "YES."
It'd be tough for me to pin down all the ways a government can impact exchange rates since most methods are fairly indirect, and frankly it's difficult to explain all the mechanics, especially in this limited space. But regardless, let me offer up a simple example that will hopefully give you a feel for government interaction in foreign exchange.
U.S. Congress Carries a Big Stick, Pressures the Japanese Yen to Rise
Companies in Japan are running into problems with their swooning local currency. This alone has caused foreign goods to become expensive, and thus the costs of imports into Japan have risen sharply. And just imagine how badly rising input costs like the price of crude oil have hit some Japanese companies.
Exporters have done well as their goods have become cheaper to foreign countries, but now they're at a point when further yen weakness isn't going to help much more. Overall it's not surprising to see Japanese businessmen lobby for a stronger yen.

Interestingly, it's not just isolated parties in Japan who are riled up over this situation. A bill was recently introduced in the U.S. Congress addressing the growing concern over a dramatically undervalued Japanese yen.
Congress has deemed this bill the "Japan Currency Manipulation Act." The men and women in the House of Representatives point towards excess U.S. dollar reserves as pressuring the yen to its current extreme. After all, inflows of dollars are ultimately balanced by outflows of yen - pushing the price of yen down versus the buck.
As procedures are discussed to reverse the extreme dollar/yen exchange rate, it will create a sense of urgency to shift the existing currency flows in Japan, even if no legal measures are eventually passed. This ongoing debate seems to have already sparked a move in the yen - a move that could easily snowball and quickly counteract some of the current imbalances.
U.S. officials have already engaged in a similar currency manipulation campaign against China, a lot like this recent bill targeting the yen. They're accusing China of artificially suppressing the value of the Yuan. The U.S. aim: To enact protectionist measures against China that would allow the Yuan to appreciate and help reinstate the competitiveness of U.S. exporters.
Pressure like that from bigmouths in Washington has swayed Chinese officials to tighten up on monetary controls. The Chinese Yuan has stepped higher as a reaction.
The same can likely be expected from Congress' new role in setting straight the Japanese yen. If this movement gains any traction, you can expect the momentum to immediately favor the Japanese yen.
And if the yen really surges, that's going to unleash huge profit opportunities across the foreign exchange markets. My regular readers and valuable premium service subscribers know exactly what I'm talking about - and the big potential profits that should follow such a big currency move.