The most cost effective mirrors to cover a large fraction of the Earth and reflect back the sun's heat are... the polar ice caps. Which we are melting because of climate change.

This is one of the main reasons people are so worried about it being "too late" to fix climate change. There are positive feedback loops like this where a little increased heat causes even more rise in temperature.

Practically, high albedo materials (essentially they reflect a lot of light but aren't necessarily mirrors because they do scatter the direction of that light) can reduce building's HVAC energy consumption. They can also effect city level microclimate (mitigating the heat island effect).

It takes a lot of high albedo area to effect the overall climate. For example, melting ice caps do decrease the albedo of parts of the Earth, accelerating climate change. Large scale increases to Earth's albedo in the form of space mirrors were seriously considered as options for global warming mitigation as late as the Obama administration. However, the risks of unintended modifications to the environment (e.g. crop failures, unpredictable weather) are generally considered too great. At present, space mirrors, with an estimated lifetime cost of $5 trillion for 50 years of cooling, could be a decent "last resort" option compared to the possibility of spending hundreds of trillions upfront to permanently solve the problem with carbon capture (although cost estimates vary widely based on the amount of CO2 that needs to be captured).

Other options include stratospheric aerosol injection (https://en.wikipedia.org/wiki/Stratospheric_aerosol_injectio...) which could minimize the effects of climate change for as little as $55 billion/year, and marine cloud brightening (https://en.wikipedia.org/wiki/Marine_cloud_brightening), which could cost as little as $5 billion/year. Possible side effects might include ozone depletion and acid rain for aerosols. Marine cloud brightening could cause weather changes.

So yes, you could manipulate albedo but it will always come with side effects and costs.

Some of the side effects are common to all kinds of solar radiation management (SRM) - geoengineering that makes the Earth more reflective in some way or another.

Greenhouse gases absorb infrared radiation emitted by the Earth and heat the atmosphere as well as the surface. SRM geoengineering just changes how much energy the Earth absorbs from the sun (which has the strongest effect at the surface).

If GHGs increase, but you use SRM to keep the surface temperature fixed, you still end up heating the atmosphere. This means in a geoengineered world, you have less rain.

A reduction in rain might be more important to some countries than others - not everyone would want the same amount of geoengineering.