I agree that 'too' is usually added. That's how I always say it and hear it. That's a minor point, in my opinion. I don't want to reinvent the wheel so here is an excellent explanation of this idiom/proverb from our friends at Wikipedia: 

You can't have your cake and eat it (too) is a popular English idiomatic <a href="https://en.wikipedia.org/wiki/Proverb" title="Proverb" style="color: rgb(11, 0, 128); background-image: none; background-position: initial; background-size: initial; background-repeat: initial; background-attachment: initial; background-origin: initial; background-clip: initial;">proverb</a> or figure of speech.<sup id="cite_ref-1" class="reference" style="line-height: 1; unicode-bidi: isolate; white-space: nowrap; font-size: 11.2px;"><a href="https://en.wikipedia.org/wiki/You_can%27t_have_your_cake_and_eat_it#cite_note-1" style="color: rgb(11, 0, 128); background-image: none; background-position: initial; background-size: initial; background-repeat: initial; background-attachment: initial; background-origin: initial; background-clip: initial;">[1]</a></sup> The proverb literally means "you cannot simultaneously retain your cake and eat it". Once the cake is eaten, it is gone. It can be used to say that one cannot or should not have or want more than one deserves or is reasonable, or that one cannot or should not try to have two incompatible things. The proverb's meaning is similar to the phrases "you can't have it both ways" and "you can't have the best of both worlds."

The first one seems true and by adding too is better: ''You can't have your cake and eat it too''...maybe too is eliminated for simplicity...It is like a famous idiom in Persian: هم خدا را خواستن و هم خرما را (wanting both God and the sugar-dates)

Idioms, like phrasal verbs, are just little chunks of speech that can't be treated logically. As Ali notes, I've always heard the phrase with the word "too" at the end: "You can't have your cake, and eat it too."

The words are what they are and mean what they mean, and the word order is traditional.

Another example, one that's become current in the last twenty years or so, is "a steep learning curve." Logically, a steep learning curve means the amount you learn rises quickly in a short period of time, so it should mean "easy to learn," but, of course, it means "hard to learn." Logical or not, if someone says "steep learning curve" they mean "difficult," and you'd better say it the same way if you want people to understand.

Another example: if something is a real bargain, people will say "it's cheap at half the price" when, logically, it should be "cheap at twice the price" (it's so cheap that you could double the price and it would still be cheap).

Try not to let it bother you.