I was writing a check and while I was writing, I spelled out value of say $124.12. I accidentally wrote "One Hundred Twenty Four and Twelve Cents", instead of "One Hundred Twenty Four and 12/100". Is there a proper way to correct this, or can I leave it as it is?
You shouldn't have a problem. I've done this in the past (in the US) and I've never had anyone at a bank complain. I recall seeing instructions that state to write cents in numeric form, e.g. 99/100, instead of word form, but I think this is merely standard practice and not required per se.
As a test, I paid a medical bill with a check written this way yesterday and asked the receptionist about it; she recalled "a good number" of checks written that way over the years. I think it's just a matter of personal preference. Although the experience of one office isn't a substitute for a large survey or some other scientific method, it's anecdotal evidence that it probably won't cause an issue for you. Just make sure you're handwriting is legible.
If you do choose to alter the check, either initial it or shred it and write a new check. As a side note, make sure that whatever form you choose to use for the amount you write out, make sure it matches the amount specified in the dollar box. The amount in the dollar box isn't crucial, since the bank will use the written amount to determine the check's value. However, it might raise eyebrows if the amounts are different and may confuse the recipient of the check.