Haskell, 163 bytes

o f=foldl1 f.concat
r=[0..3]
q n=take(min(n+2)3).drop(n-1)
0#m=m
v#m=[o max$q y$q x<$>n|y<-r,x<-r,m!!y!!x==v]!!0#n where n=map(z<$>)m;z w|w==v=0|0<1=w
f=o(+).(16#)

Try it online!

The f function takes the input as a list of 4 lists of 4 integers.

###Slightly ungolfed

-- helper to fold over the matrix
o f = foldl1 f . concat

-- range of indices
r = [0 .. 3]

-- slice a list (take the neighborhood of a given coordinate)
-- first we drop everything before the neighborhood and then take the neighborhood itself
q n = take (min (n + 2) 3) . drop (n - 1)

-- a step function
0 # m = m -- if the max value of the previous step is zero, return the map
v # m = 
    -- abuse list comprehension to find the current value in the map
    -- convert the found value to its neighborhood,
    -- then calculate the max cell value in it
    -- and finally take the head of the resulting list
    [ o max (q y (q x<$>n)) | y <- r, x <- r, m!!y!!x == v] !! 0 
       # n -- recurse with our new current value and new map
    where 
        -- a new map with the zero put in place of the value the mouse currently sits on 
        n = map (zero <$>) m
        -- this function returns zero if its argument is equal to v
        -- and original argument value otherwise
        zero w 
            | w == v = 0
            | otherwise = w

-- THE function. first apply the step function to incoming map,
-- then compute sum of its cells
f = o (+) . (16 #)

Haskell, 163 bytes

o f=foldl1 f.concat
r=[0..3]
q n=take(min(n+2)3).drop(n-1)
0#m=m
v#m=[o max$q y$q x<$>n|y<-r,x<-r,m!!y!!x==v]!!0#n where n=map(z<$>)m;z w|w==v=0|0<1=w
f=o(+).(16#)

Try it online!

The f function takes the input as a list of 4 lists of 4 integers.

Slightly ungolfed

-- helper to fold over the matrix
o f = foldl1 f . concat

-- range of indices
r = [0 .. 3]

-- slice a list (take the neighborhood of a given coordinate)
-- first we drop everything before the neighborhood and then take the neighborhood itself
q n = take (min (n + 2) 3) . drop (n - 1)

-- a step function
0 # m = m -- if the max value of the previous step is zero, return the map
v # m = 
    -- abuse list comprehension to find the current value in the map
    -- convert the found value to its neighborhood,
    -- then calculate the max cell value in it
    -- and finally take the head of the resulting list
    [ o max (q y (q x<$>n)) | y <- r, x <- r, m!!y!!x == v] !! 0 
       # n -- recurse with our new current value and new map
    where 
        -- a new map with the zero put in place of the value the mouse currently sits on 
        n = map (zero <$>) m
        -- this function returns zero if its argument is equal to v
        -- and original argument value otherwise
        zero w 
            | w == v = 0
            | otherwise = w

-- THE function. first apply the step function to incoming map,
-- then compute sum of its cells
f = o (+) . (16 #)