Is it possible to recover from a single engine failure at cruising altitude on the HondaJet and safely land the aircraft?
There would not be much point in having two engines if a single engine failure would be unrecoverable. In fact, that would make the plane twice as likely to have an unrecoverable event!
If one engine has a 1/1million chance of failure, then the chance of failure of at least one of 2 engines is 2-in-a-million.
(technically, slightly less than 2-in-a-million... but close enough)
And if either engine failure is catastrophic, then you've doubled your chances of losing the plane.
I don't know about the HondaJet in particular, but by logic and statistics, I can say that a single engine failure must be recoverable.
The HondaJet is type certified by the FAA. The type certificate says that it's certified in the normal category under 14 CFR 23, which includes the requirements to be able to take off and climb on a single engine.
Simply put, the HondaJet has plenty of surplus thrust even on one engine and there should be no problem at all landing it after a single engine failure, assuming no other complications.
While the others answers are completely correct regarding the ability of the Hondajet (and any other twin jet, for that matter) to safely fly after an engine failure (including during the takeoff roll past V1,) it's probably also worth pointing out that you don't need any engines to safely land from cruising altitude.
Engines are generally needed to climb or maintain altitude, but you can glide with no engines at all. According to Honda's website, the Hondajet has a maximum cruise altitude of 43,000 ft and has a cruise speed of 420 KTAS if cruising at 30,000 ft. Even if you lost both engines at 30,000 ft., you'd almost certainly be able to glide around a hundred miles and still land safely. From 43,000 ft. it would probably be well over 100 miles. This would be true for almost any commercial or private jet aircraft.
Normally, for a dual-engine aircraft to be certifiable, it must be able to fly on a single engine.
In practice, a single engine failure is very dangerous because the dead engine generates huge drag, so the yaw effect is enormous. In many cases the pilot must reduce the power from the remaining engine to mitigate yaw imbalance, so the aircraft ends up flying at a minimal power which is dangerous. Nevertheless, in most cases if the failure occurs at altitude, there is a good likelihood of being able to land successfully if good flying and procedures are followed.
There is an old joke, "the purpose of the second engine is to fly you to the site of the crash in case of an engine failure".
Contrary to what another answer states, dual engines are not used to provide safety redundancy, they are used to provide more power. As a general rule, aircraft designs with dual engines will usually have greater power, speed and range than comparable single-engine designs. Dual engines are actually more dangerous than single engines because they are more complex to fly and you have double the chances of engine failure.