I've been thinking about entering some writing contests. I fully understand that these contests, in order to protect their rights to the first-place pieces, often request you withdraw your piece from submission circulation until their winners are announced. For some competitions, that can mean a wait of 6, 7, or 8 months from the time you submit until the announcement date.
While I believe my work is good enough to be entered and noticed, I honestly don't believe I will be the ONE first-place winner in a national contest with hundreds, maybe thousands of entrants. I'd more likely find a living dinosaur grazing peacefully in my backyard! What I'm hoping to do with my entries is garner at least some type of recognition: an encouraging response, a contact with a possible publisher/editor, or maybe an honorable mention or two. Those things look good on cover letters. [They look good to you and to your friends and family, and maybe on a job resume at a PR firm, but agents and publishers mostly ignore this.]
If I do enter these contests, should I continue to submit my pieces through the standard, often slow editorial process during the competition's 'waiting' period? And if that is okay, should I mention to the other publishers that this piece is currently an entrant in The XYZ Writing Competition to be decided on such-and-such a date? What do you recommend?
The first thing you need to do is determine why you want to enter the contest. If it's not to win, why bother? There are faster and easier ways to get encouraging responses and attention, like a writers critique group. There are writer associations that hold annual conferences that often feature critique as part of their event. (Ex: LDS Storymakers)
I, personally, would not remove a piece from submission to enter it into a contest. If it's good enough to garner contest attention, then it will also catch the attention of the publisher--and that's your end goal, right? Why back away from it?
If you have a piece that you're not currently submitting and you want to put it in a contest, that's fine. Make sure the contest is legit--there are judges with credentials (a variety of professional editors, writers, etc.), prizes that are meaningful (cash, publication, etc.) and the entry fee is reasonable (under $100). You can find out about scam contests at Preditors & Editors.
I don't care what contest you're entered in. I don't usually care if you've won. Don't mention that you've entered a contest in your query unless it will affect the agent's/editor's ability to acquire it. And if it will, then don't submit it. If you happen to win a legit contest, you may put that at the end of your query if you feel you must, but quite honestly, the quality of your writing is what is going to sell your piece, not any contest you may have won.