You are seeing this effect because LiDAR doesn't like water, it acts like a mirror, in the centre of a run (flight) the intensity is very high but just outside centre the pulse is reflected away (angle of incidence = angle of reflection) and does not return to the sensor and therefore records no return. You can interpolate with a terrain, provided you have 3d analyst extension available or with a LAS dataset (basic license requires 3d or spatial analyst).
Interpolating will close in gaps between valid records - ALL gaps up to a boundary called a convex hull, even over the land. You could make a terrain or LAS dataset from all your records restricted to ASPRS class 2, 8 and 9 (Ground, Model Key and Water), convert to raster and then Extract by Attributes to restrict to elevations below your intended area.
To import your LAS for a terrain use LAS to MultiPoint which is far more efficient than creating single points for each pulse/return. (3d analyst required)
To export a terrain to a raster you will need 3d analyst regardless of license level, use the tool Terrain to Raster. Linear interpolation method is faster but can look blocky, Natural Neighbors is slower but creates a smoother raster.
To export a LiDAR (LAS) dataset to raster use the LAS Dataset to Raster tool which will offer more interpolation methods but as you want to fill in the blanks stick with Natural Neighbors or Linear fill method.
If you are on a basic license with no 3d or spatial analyst extension you can look at LASTools as a viable alternative, there's a GUI available in QGIS. Licensing isn't too expensive and the tools are efficient; Or contact your local Esri representative to obtain extension(s) or upgrade your license.