When having your cataracts removed, can you have a lens put in to improve your vision?

Yes, that is routine. A cataract is the clouding of the eye's natural lens. Virtually all cataract surgery involves removing the cloudy lens and placing an artificial intra-ocular lens (iol) implant in its place. There are some newer more advanced iols that act like intraocular invisible bifocals to minimize need for postoperative glasses (multifocal iols) or to treat pre-exisiting astigmatism (toric iol).
Absolutely. It is one of the benefits of surgery and should be discussed with your surgeon.