How are energy and matter added to and removed from the Earth system?

energy is traveling through space as "light". Most of the energy earth gets from outside of the earth comes from the sun. The gravitational pull of outside objects is also a minor source of energy within the earth (takes energy to make the tides work and that energy isn't from inside earth, and some of that energy converts into heat from friction).

The earth emits energy primarily as heat (infrared light). some of the light energy coming in from outside is also reflected back outside, not always in the same wavelengths that it comes in as.

There is also kinetic energy coming into earth with particulates (anything from asteroids down to ionized atoms).

This last group of stuff is the main form of matter (mass) addition. something like a million tonnes of debris enters the earth per year, give or take a couple orders of magnitude depending on who you believe.

Matter is lost primarily from the very upper atmosphere where the solar wind strips away matter, mostly hydrogen atoms although larger stuff also escapes gravity. The amount of mass that is lost to space is a couple orders of magnitude (something like 1000 times) less than the amount that is gained per year. I suppose there is some kinetic energy lost from earth as this mass escapes, just as there is some energy gained when stuff falls in.

The earth is much more open to energy than to mass. That is, the earth receives and emits significant energy, enough that the temperature of the outer zone, the atmosphere and oceans for the most part, is way warmer than it would be without outside inputs, but the amount of mass gained and lost is trivial (way way tiny) compared to the total mass of the earth. the mass of the earth really doesn't much change even over millions of years; the temperature (energy) of the air-water system is always changing (we think of that change as climate) even over decades and centuries).

If you had ever been to a desert, you would know how efficiently the planet radiates heat away: temperatures drop 30 degrees Celsius over night due to nothing else but radiation. The only reason this does not happen nearly as fast over non-arid land areas is the water vapor in the air which creates a significant greenhouse effect. Take it away, e.g. in winter, and the nights are getting cold VERY rapidly. In other words: your question starts with the wrong assumption. There is not much too it otherwise. And we do not need space mirrors. We are getting plenty of solar radiation the natural way.

"The atmosphere constantly interacts with other components of the Earth system.

Atmospheric processes interact and transfer energy and matter with other components of the Earth system such as: water, ice, land, etc.. For example, the exchanging of energy between oceans and atmosphere in hurricane formation. "

Energy: One way is radiation added by the sun. Loss is radiation lost from the surface facing away from the sun.

Matter: Gained by space debree being sucked in by gravity. Lost by .... space craft and rockets blasting it into space. .