'earnestly ask' is correct and usable in written English.
You can use it to emphasize the importance and sincerity of a request or question. For example: I earnestly ask that you respond to this message as soon as possible.
On Wednesday, the group added, "We earnestly ask the Pakistani government and officials to give clarification, and, just as they have announced, he should be released".
I earnestly ask you to do all you can to help maintain the superb research at the Koch Institute at its maximum level".
How does he listen to this sophist and then earnestly ask if he can, "with right and conscience," make a claim to the French throne?
In the middle of press conferences, reporters would earnestly ask him about solving the Palestine problem, or if he could have a quiet word with Moscow about President Ronald Reagan's star wars programme.
The development was not unexpected a campaign known as Youths Earnestly Ask for Abacha was recently given several million dollars' worth of state money to hold a rally but it makes a mockery of the general's 1994 pledge to return the country to democratic rule "in which we [the armed forces] are not participants".The decision illustrates three truths about Nigeria.
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One earnestly asked me to explain my opinion about the state of things.
"How do I act so well?" Ian McKellen earnestly asks Mr. Gervais's character in a forthcoming episode.
"Do you know anything about psychoanalysis?" Jerry earnestly asks him, and the line gets an immediate laugh.
As a high school student during the psychedelic Sixties, he earnestly asked classmates to focus long enough to become Young Republicans.
It is easy to see why the film's distributors "are earnestly asking audiences not to reveal how it ends," Bosley Crowther wrote in The New York Times.
