The Competitive Cue bid is a strange unnatural bid I'll grant you. But it is hugely valuable, especially when both sides bid. In these situations a cue-bid is defined as "a bid of the suit opponents have bid". It cannot possibly be "natural". That is to say, when you do bid the opponents suit you cannot possibly be interested in playing in that suit. If you do - genuinely - hold their suit, you should either be passing to begin with, or later on in the auction, doubling them for a big fat juicy penalty. The last thing you want is to play in THEIR suit. It follows therefore that, when you bid their suit, it must be for another reason. There are two distinct and separate situations when this can occur. Partner will (1) open a minor suit or (2) open a major suit.
(1) when partner opens a minor suit we will - most of the time - be aiming to play in a NT contract. 9 tricks are easier than 11. After an overcall on your right (1D - 1S!) you may want to play in NT but have no "stop" (ie high honour) in their suit but still have a lot of points. Don't "guess" 3NT - use the cue-bid to ask partner if they have a "stop"
You bid their suit as in 1D - 1S - 2S! This says "do you have a stop in their suit such that you can suggest a NT contract?" If possible partner will duly bid NT at some level or other depending on their hand. (Note that with 4 H's this hand would use the "Negative" double!)
2) When partner opens a major suit we will - most of the time - be looking to express a fit for partner and aiming for game. After an overcall on your right, you now want to bid the value of your hand either pre-emptively (weak) or constructively (strong). But how can we distinguish the difference? After 1H - 1S (overcall) what do you do with this:
You must make a weak (!) pre-emptive jump to 3H! Make things difficult for them. Push them into guessing what to do next. Obstruct. If you do then sometimes they will get it wrong. Conversely what do you do with this?
After 1H - 1S (overcall) you have a good fit for partner (gold dust) as well as good (10+) points (platinum dust). You must show that your raise is strong and NOT pre-emptive. You do this by bidding their suit thus:
1H - 1S - 2S!