It seems that your system has not enough equations:
w[h_[t]] := h[t] - (1 - h[t])*g[t]
dgl = {
h'[t] == y[t]*(1 - h[t])*g[t] - a*h[t],
y'[t] == y[t]*b*(1 - c)*w[h[t]] - y[t]*d*(x[t] + y[t]),
g'[t] == V*D[w[g[t]], t]}
NDSolve[dgl /. {a -> .1, b -> .6, c -> .03, d -> .7, V -> 2.3},
{x, y, g, h},
{t, 0, 5}]
You need another one for x. So with some (arbitrary, you should give them according to your problem) modifications the system runs and you get an answer
dgl = {
h'[t] == y[t]*(1 - h[t])*g[t] - a*h[t],
y'[t] == y[t]*b*(1 - c)*w[h[t]] - y[t]*d*(x[t] + y[t]),
g'[t] == V*D[w[g[t]], t],
x'[t] == g[t],
x[0] == 0,
y[0] == 0,
g[0] == 0,
h[0] == 0
}
NDSolve[dgl /. {a -> .1, b -> .6, c -> .03, d -> .7, V -> 2.3},
{x, y, g, h},
{t, 0, 5}]