It is a grass but I have no idea which one. Could you just mow it along with the rest of the lawn? It's only flowering because the lawn has got so long.

Looks like annual bluegrass to me.

We mow the lawn once in 2 weeks. But within a couple is days this comes back. Will weed b gone work?

Since it is a grass any weedkiller that kills it will also kill your 'official' grass. Can you mow more often? Once a fortnight doesn't seem very often. I don't know your climate but quite possibly it will languish if the summer is hot where you are.

Whoever told you your lawn can only have 1 kind of grass was trying to sell some chemical or other grass seed, possibly with a faux Scottish accent.

Whether it's annual or perennial, waiting until it develops (and drops) ripe seeds is not a good idea if you can't tolerate it.

If you must have one kind of grass and nothing else, you should know what grass you have and provide the conditions for making it thrive. Hopefully the climate isn't the main negative factor.

It looks like annual bluegrass to me. I don't have a faux Scottish accent and am not trying to sell anything, but if it IS annual bluegrass products with Dimensionî pre-emergent can control it.

Could be a Rye grass if it looks like miniature wheat sprouts.

Here is a link that might be useful: rye grass