Jelly, 8 bytes

ɓḟŒP§iµƇ

Try it online!

Essentially, we go over each element, x, of the input, L, and keep it if:

• We remove x from the input, call that L'
• We then get all subsets of L', and get the sums of each
• Finally, if x is in these sums, we keep it

If 0 is not considered the sum of the empty list, then +1 byte

How it works

ɓḟŒP§iµƇ - Main link. Takes a list L on the left
ɓ     µ  - Group the links between ɓ and µ into a dyad f(L, x):
 ḟ       -   Remove x from L, L'
  ŒP     -   Powerset of L'
    §    -   Sums of each
     i   -   Index of x, or 0 if not present
       Ƈ - For each element, x, of L, keep it if f(L, x) is true

Jelly, 9 bytes

ɓḟŒPḊ§iµƇ

Try it online!

Essentially, we go over each element, x, of the input, L, and keep it if:

• We remove x from the input, call that L'
• We then get all non-empty subsets of L', and get the sums of each
• Finally, if x is in these sums, we keep it

If 0 is not considered the sum of the empty list, then +1 byte

How it works

ɓḟŒPḊ§iµƇ - Main link. Takes a list L on the left
ɓ      µ  - Group the links between ɓ and µ into a dyad f(L, x):
 ḟ        -   Remove x from L, L'
  ŒP      -   Powerset of L'
    Ḋ     -   Remove the leading empty array
     §    -   Sums of each
      i   -   Index of x, or 0 if not present
        Ƈ - For each element, x, of L, keep it if f(L, x) is true

Jelly, 8 bytes

ḟŒP§iʋ@Ƈ

Try it online!

Essentially, we go over each element, x, of the input, L, and keep it if:

• We remove x from the input, call that L'
• We then get all subsets of L', and get the sums of each
• Finally, if x is in these sums, we keep it

How it works

ḟŒP§iʋ@Ƈ - Main link. Takes a list L on the left
       Ƈ - Filter each element of L, x, on the following:
     ʋ@  -   Group the previous 4 links into a dyad f(L, x):
ḟ        -     Remove x from L, L'
 ŒP      -     Powerset of L'
   §     -     Sums of each
    i    -     Index of x, or 0 if not present

Jelly, 8 bytes

ɓḟŒP§iµƇ

Try it online!

Essentially, we go over each element, x, of the input, L, and keep it if:

• We remove x from the input, call that L'
• We then get all subsets of L', and get the sums of each
• Finally, if x is in these sums, we keep it

If 0 is not considered the sum of the empty list, then +1 byte

How it works

ɓḟŒP§iµƇ - Main link. Takes a list L on the left
ɓ     µ  - Group the links between ɓ and µ into a dyad f(L, x):
 ḟ       -   Remove x from L, L'
  ŒP     -   Powerset of L'
    §    -   Sums of each
     i   -   Index of x, or 0 if not present
       Ƈ - For each element, x, of L, keep it if f(L, x) is true

Jelly, 8 bytes

ḟŒP§iʋ@Ƈ

Try it online!

How it works

ḟŒP§iʋ@Ƈ - Main link. Takes a list L on the left
       Ƈ - Filter each element of L, x, on the following:
     ʋ@  -   Group the previous 4 links into a dyad f(L, x):
ḟ        -     Remove x from L, L'
 ŒP      -     Powerset of L'
   §     -     Sums of each
    i    -     Index of x, or 0 if not present

Jelly, 8 bytes

ḟŒP§iʋ@Ƈ

Try it online!

Essentially, we go over each element, x, of the input, L, and keep it if:

• We remove x from the input, call that L'
• We then get all subsets of L', and get the sums of each
• Finally, if x is in these sums, we keep it

How it works

ḟŒP§iʋ@Ƈ - Main link. Takes a list L on the left
       Ƈ - Filter each element of L, x, on the following:
     ʋ@  -   Group the previous 4 links into a dyad f(L, x):
ḟ        -     Remove x from L, L'
 ŒP      -     Powerset of L'
   §     -     Sums of each
    i    -     Index of x, or 0 if not present