"eagerly hoping" is correct and usable in written English.
You can use it when you want to express that someone is excitedly looking forward to something. Example: She was eagerly hoping to hear the news about the job.
I have seen families gathered around the U.S. women's national team training field in Scottsdale, Ariz., eagerly hoping to get an autograph.
However, for Back to the Future fans eagerly hoping to get their hands on the iconic car, it will not come cheap.
Never again will I check my mobile inbox eagerly, hoping for a text from that special someone, only to find this: "Ab motapa ghatana bilkul aasan, bina vyayam, bina upwas, bina yoga ke, motapa ghataye, warna paise wapas paye".
Those hired to guide you through the unknown or to cleverly enhance what you already know are awaiting you eagerly, hoping you will give them the chance to do so.
With an air of glamour maintained by media like these, many young women gravitate to Kabukicho entirely of their own accord, eagerly hoping for what they dreamily conceive of as a life of adventure and even celebrity.
That was because the influential group of Republicans eagerly hoping to gain control of the White House and the nominating procedure has adopted "No More Souters" as its private slogan, one G.O.P. adviser said.
Instead, the president eagerly hoped that the state might voluntarily emancipate its slaves.
With this in mind, we hope Alice has managed to feel at home in Norwich like she had eagerly hoped.
It is eagerly hoped that the new parliament will be better than its woeful predecessors and will soon choose a plausible government.
"We have already discovered an astonishing amount of new information using x-ray fluorescence, and eagerly hope for more," said William Noel of The Walters Art Museum and director of the Archimedes Palimpsest collaboration.
Nevertheless, the announcement that topless bathing would be allowed brought a strong negative reaction from the Roman Catholic Church of Uruguay and, as Uruguayan businessmen eagerly hoped, a news media frenzy in Argentina.
