Answer: Without seeing the plant it is difficult to identify and to suggest a control measure. However, it sounds to me like Pearlwort, rather than a mosss or algae. Pearlwort is a perennial, spreading by seed and rooting of stems. It forms a dense mat close to the ground and the very fine, grass-like leaves can be mistaken for a moss. It is very invasive and a problem in close cut lawns. You can raise the mower to allow the grass to outgrow it.
The seed can be spread while mowing and carried on soles of shoes. It flowers from April to September, with small bead-like buds. You can hand weed, as regrowth from the roots is not a problem, but it can be difficult to remove when you have large patches. I'd recommend simply slicing it off the surface with the edge of a flat shovel. Or, you can use a weed and feed product on your lawn. It may take 2-3 applications before you get this weed under control.
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