We know that Moshe had a speech impediment. But why didn’t Moshe ask Hashem to heal him from this?
The Midrash tells us that in Paroh’s palace they brought before Moshe a golden piece and a coal and Malach Gavriel pushed his hand so that Moshe would grab the coal. Moshe then placed his hand with the coal into his mouth and his tongue became burned. From this came forth his speech impediment. Moshe knew that an open miracle happened to him.
Now, the Midrash tells us that Moshe was to say to the Jewish people פקד פקדתי, I have surely remembered you, as there was a tradition from Yosef that Hashem will redeem the Jewish people from Mitzraim with that phrase. This phrase showed the person who used these words was the true redeemer. How did this phrase prove it? The letter פ can only be said properly by using the lips. If Moshe would ask Hashem to heal him then he would have no problem saying פקד פקדתי. As a result, if every Jew knew that the one taking the Jews out will say פקד פקדתי, then anyone can do that. However, because Moshe had this blemish, resulting in his inability to say the פ and suddenly he could say it, this proved he wasn’t an imposter.