there is a gap when two line pass the same point, at x=0.3. How to eliminate the gap? Many thanks!

This is because of the hard bend near x=0.3 and the smooth option and the then "low" amount of samples where the function is evaluated. You could show that by adding markers to the "func" plots as in the given example below. To circumvent your problem you could simply increase the number of samples where the function is evaluated, e.g. to 101. (The default number of samples is 25.)

% used PGFPlots v1.17
documentclass[border=3mm]{standalone}
usepackage{pgfplots}
    usepgfplotslibrary{fillbetween}
    pgfplotsset{
        compat=newest,
        width=10cm,
    }
begin{document}
    begin{tikzpicture}[
        declare function = {
            func1(x)= (x<=0.3) * (10*x*x - 3*x ) + (x>0.3) * (x* x*x);
            func2(x)= (x<=0.3) * (0) + (x>0.3) * (x* x*x);
        },
    ]
        begin{axis}[
            ymin=-0.4,
            ymax=1,
            xmin=0,
            xmax=1,
            ytick={-0.4, -0.2, 0, 0.2, 0.4, 0.6, 0.8, 1},
            grid,
            domain=-0.4:1,
%            samples=101,   % <-- uncomment me to improve the result
        ]

            addplot [name path=linear,black] {x};
            addplot [name path=line,black] {0};
            addplot [
                name path=poly,
                blue,
                smooth,
                mark=o,
            ] {func1(x)};
            addplot [
                name path=poly2,
                red,
                smooth,
                mark=+,
            ] {func2(x)};

            addplot fill between[
                of=poly and line,
                split,
                every odd segment/.style={red!10},
                every even segment/.style={yellow!10}
            ];

            addplot [blue!10] fill between [of = poly2 and linear];

            node at (axis cs:0.18,-0.05) [anchor=north east] {A};
            node at (axis cs:0.45,0.2)   [anchor=south west] {B};
            node at (axis cs:0.8,0.3)    [anchor=south west] {C};
        end{axis}
    end{tikzpicture}
end{document}