Sum of Modulo Sums

05AB1E, 12 10 bytes

v¹¹N¹‚£O%O

Uses the CP-1252 encoding. Try it online!

JavaScript, 43 47 bytes

f=
n=>eval(n.replace(/./g,'+'+n+"%($`+ +'$&$'')"))

I.oninput=_=>O.value=f(I.value)

<input id=I>
<input id=O disabled>

Takes input as string.

Edit:

+4 bytes: Leading zeroes in JavaScript converts the number to octal ):

Brachylog, 20 bytes

:{$@~c#C:@$a+:?r%}f+

Try it online!

Explanation

This implements the formula given. The only thing we have to be careful about is when a 0 is in the middle of the input: in that case Brachylog gets pretty quirky, for example it won't accept that a list of integers starting with a 0 can be concatenated into an integer (which would require ignoring the leading 0 — this is mainly programmed that way to avoid infinite loops). Therefore to circumvent that problem, we convert the input to a string and then convert back all splitted inputs into integers.

                       Example Input: 47852

:{               }f    Find all outputs of that predicate: [716,205,152,4769]
  $@                     Integer to String: "47852"
    ~c#C                 #C is a list of two strings which when concatenated yield the Input
                           e.g. ["47","852"]. Leave choice points for all possibilities.
        :@$a             Apply String to integer: [47,852]
            +            Sum: 899
             :?r%        Input modulo that result: 205
                   +   Sum all elements of that list