Yes you are absolutely correct @srikirs007. Basically this word like many other words can be used in more than ways - one of which is countable and the other is un-countable. And this depends on the context in which the word has been used. So you always need to understand the MEANING or the INTENT of this word.
If you want to measure the extent of motivation offered in ONE particular INCENTIVE, then incentive is singular. For example your second sentence:
I have a little inventive to take up the promotion.
Another example - Not much incentive has been provided to the farmers to grow organic foods.
On the other hand, incentive is plural when you can talking about multiple incentives: Here comes your first example:
I give my kids many incentives to wake up early.
Another example - Department of Agriculture has implemented about 10 incentives to motivate farmers to grow organic food, each incentive offering both financial and social motivation.
So never just blindly look at a noun and determine its countable nature. Yes there are those words that are countable and the ones that are un-countable. But there definitely are certain words that go either way and this depends on the context in which they are used. So always pay attention to the meaning.
Let's see if I can get some examples for the word "competition". So basically - Give me two examples - 1 in which competition is used in un-countable sense and the other in which competition is used in countable sense.
I look forward to your response.
Takeaway - Go by the context of the sentence.
I have a fierce competition with myself to achieve the unexpected.