Easter Island, Machu Picchu and the Galapagos Islands?
In short, yes it can. You’ll probably want to spend one of your three weeks in Peru, followed by four nights on Easter Island, a 7-night cruise in the Galápagos and three positioning nights (to make sure everything runs smoothly and at a comfortable pace) in Ecuador.
The quickest way to get between Easter Island and Machu Picchu will be to fly via Lima and then Chile’s capital Santiago. Either way three weeks is plenty of time to visit all three highlights and it will also allow time for you to acclimatise to the altitude in Peru, which is important when visiting Cusco and Machu Picchu.
There are also direct flights connecting both Lima and Santiago to Ecuador, from where you can start a 7-night cruise (or a shorter one if you prefer) around the Galápagos Islands.
You could visit Easter Island, Machu Picchu and the Galápagos Islands in any order but most people prefer to end with a cruise as it is a relaxing way to wind down towards the end of a trip.
I would not recommend you include Guyana (which is perfectly safe when travelling with a tour operator) in the same holiday. Flights from the capital Georgetown to Peru, Santiago or Ecuador will not be very convenient – most likely you will need to fly via Port of Spain, Trinidad, and then change planes in Venezuela or Colombia. The cost of the flight is likely to add £400-£500 to your trip. Three days in Guyana will not be enough either I'm afraid as the main highlights are located fairly deep in the country's interior, so you’d be looking at a road transfer of 5-6 hours from Georgetown on the coast before reaching one of the lodges. If you are keen to include a jungle trip, though, you could certainly consider the Peruvian Amazon. You can easily fly from Cusco to Puerto Maldonado in the Amazon and in no time at all be fishing for piranhas!