here's a close up. i think at some point it bears a small red "fruit" similar to a strawberry if i'm not mistaken.

Are you sure it is this plant which has the red fruit and not another similar plant? This one looks like a Potentilla aka Cinquefoil. The slightly similar one with red fruit is Duchesnia indica. It's possible you have both.

As to destruction either you dig it out or you use a plant poison. But personally I can't see the point in removing something green and attractive from a fairly naturalistic, rough-looking lawn.

Wild, or Alpine, Strawberry. The berries are edible and are often much sweeter then commercially grown strawberries.
Like wild Violets I have not found anything that will "destroy" these short of killing off the grass they are growing in. Keep the runners they send out to spread cut to contain them.

Here is a link that might be useful: wild strawberry

Kimmsr, yours 'strawberry' has three leaflets, not five.

Floral is correct with the ID of Cinquefoil, which also has small, red berries.

Now that's interesting rhizo_1. Which species were you thinking about? I couldn't think of any Cinquefoils that had red berries. That's why I suggested there might be some Duchesnia in the area too. But now I find our native wild strawberry has been reclassified in the Potentillas. But the OPs plant is not a strawberry so the red fruit remain a mystery to me.

Here is a link that might be useful: Fragaria/Potentilla