In this post ,I observed computationally that the mock theta functions of order $3$,found in this wikipedia article
$f(q)=\sum_{n=0}^{\infty} \frac{q^{n^2}}{(-q;q)^2_{n}}$,$\phi(q)=\sum_{n=0}^{\infty} \frac{q^{n^2}}{(-q^2;q^2)_{n}}$,$\omega(q)=\sum_{n=0}^{\infty} \frac{q^{2n(n+1)}}{(q;q^2)^2_{n+1}}$ and $\nu(q)=\sum_{n=0}^{\infty} \frac{q^{n(n+1)}}{(-q;q^2)_{n+1}}$
where $q=e^{2\pi i\tau}$,$|q|\lt1$ and $(a;q)_n=\prod_{j=0}^{n-1}(1-aq^j)$
satisfy the following relations
$\Big(\phi(q)-1\Big)^2+\Big(f(q^2)-1\Big)=2\sum_{k=1}^{\infty}\Big(\sum_{n=1}^{\infty} \frac{q^{(n-1)^2+2k(n-1)+2k^2}}{(-q^2;q^2)_{k} (-q^2;q^2)_{n+k-1}}\Big)\tag1$
and
$\Big(\nu(q)-\frac{1}{(1+q)}\Big)^2+\Big(\omega(-q)-\frac{1}{(1+q)^2}\Big)=2\sum_{k=1}^{\infty}\Big(\sum_{n=1}^{\infty} \frac{q^{n^2+(2k-1)n+2k^2}}{(-q;q^2)_{k+1} (-q;q^2)_{n+k}}\Big)\tag2$
and fellow user @Somos answered that the identities are special cases of the general identity
$S_1^2+S_2=2\sum_{i\le j}a_ia_j$
resulting from the theory of symmetric polynomials
But the natural question that arises in the context of modular forms is
since the RHS of $(1)$ and $(2)$ is related to the mock theta functions by the relations,does it satisfy certain modular properties like mock theta functions?
Rearrange relation $(1)$ and let $q\rightarrow q^{1/2}$
$f(q)=2\sum_{k=1}^{\infty}\Big(\sum_{n=1}^{\infty} \frac{q^{((n-1)^2+2k(n-1)+2k^2)/2}}{(-q;q)_{k} (-q;q)_{n+k-1}}\Big)-\phi^2(q^{1/2})+2\phi(q^{1/2})\tag{3}$
and similarly rearrange $(2)$ and let $q\rightarrow -q^{2}_{1}$
$\omega(q^2_1)=\nu^2(-q^2_1)-\frac{2}{(1-q^2_1)}\nu(-q^2_1)-2\sum_{k=1}^{\infty}\Big(\sum_{n=1}^{\infty} \frac{q^{2n^2+(2k-1)2n+4k^2}_1}{(q^2_1;q^4_1)_{k+1} (q^2_1;q^4_1)_{n+k}}\Big)\tag{4}$
after substituting relation (3) and (4) into
$q^{-1/24}f(q)=2y^{-1/2}q^{4/3}_1\omega(q^2_1)+4(3y)^{1/2}\int_{0}^{\infty}e^{-3\pi yx^2}\frac{\sinh{2\pi yx}}{\sinh{3\pi yx}}\,dx\tag{5}$
mentioned in the comments due to Ramanujan and later rediscovered by Watson(see [1]),
we are led to
$2y^{-1/2}q^{4/3}_1\Big(\nu^2(-q^2_1)-\frac{2}{(1-q^2_1)}\nu(-q^2_1)-2\sum_{k=1}^{\infty}\Big(\sum_{n=1}^{\infty} \frac{q^{2n^2+(2k-1)2n+4k^2}_1}{(q^2_1;q^4_1)_{k+1} (q^2_1;q^4_1)_{n+k}}\Big)\Big) +4(3y)^{1/2}\int_{0}^{\infty}e^{-3\pi yx^2}\frac{\sinh{2\pi yx}}{\sinh{3\pi yx}}\,dx=q^{-1/24}\Big(2\sum_{k=1}^{\infty}\Big(\sum_{n=1}^{\infty} \frac{q^{((n-1)^2+2k(n-1)+2k^2)/2}}{(-q;q)_{k} (-q;q)_{n+k-1}}\Big)-\phi^2(q^{1/2})+2\phi(q^{1/2})\Big)\tag{6}$
where $q=e^{-2\pi y}$ and $q_1=e^{-\frac{\pi}{2y}}$
and we see that the mock theta functions $\nu(q)$ and $\phi(q)$ are related under modular transformation and also the summations as well.
References
[1]:W Duke:Almost a Century of Answering the Question:What is a Mock Theta Function? August 11 2014