Pseudo Goldstone bosons are exceptionally light scalars in systems with both spontaneous and explicit symmetry breaking, the latter serving as a small perturbation to the former.
I'll vulgarize the standard SU(2) σ-model of pions, the original pseudogoldstons, serving as the prototype of all that followed, the serious student should find, in, e.g., Itzykson & Zuber, Section 11-4.
To this end, I'll just keep one σ and one π, rotating into each other in a plain O(2), the real field presentation of U(1), and skip kinetic and Yukawa terms to focus on the celebrated Goldstone sombrero potential (1961). In suitable over-all normalizations,
$$
V_0= \frac{\lambda}{4} (\sigma^2 + \pi^2 -v^2)^2,
$$
which is manifestly invariant under field plane rotations,
$$
\delta \pi = \sigma, \qquad \delta \sigma = -\pi~.
$$
The standard SSB choice dictates that, at the minimum,
$$
\langle \sigma\rangle=v ~, \qquad \langle \pi\rangle=0~, ~\Longrightarrow \langle \delta \pi\rangle=v,
$$
identifying the π as a goldston. Further defining $\sigma'\equiv \sigma -v$, so that $\langle \sigma'\rangle=0$, it is evident the potential expands to interaction terms, no π mass term and a σ' mass term with $m_{\sigma'}^2= \lambda v^2$, so the σ' is a dull conventional field. This is plain SSB.
Suppose now the gods of current quark masses break the symmetry explicitly by a small amount proportional to the dimensionless parameter $\epsilon$, a small perturbation whose $O(\epsilon^2)$ expressions we'll systematically discard. That is, they tilt the sombrero slightly in the σ direction, favoring the original vacuum,
$$
V_\epsilon= \frac{\lambda}{4} (\sigma^2 + \pi^2 -v^2)^2 -\bbox[yellow,5px]{\epsilon \lambda v^3 \sigma }~.
$$
It is evident the above rotation transformation does not leave the perturbation invariant, so the original O(2) symmetry is explicitly broken to $O(\epsilon)$ in the potential, on top of the huge SSB.
To lowest order in $\epsilon$, the minimum of the potential is at $\langle \pi\rangle = O(\epsilon)$, with mostly ignorable effects, and
$$
\langle \sigma\rangle (\langle \sigma ^2\rangle -v^2) -\epsilon v^3=0 ~~~\Longrightarrow ~~~ \langle \sigma\rangle = v(1+\bbox[yellow,5px]{\epsilon /2}).
$$
Again, defining $\sigma ''\equiv \sigma -v(1+\epsilon /2)$ and not bothering with shifting π, since such a shift would not result in order $O(\epsilon)$ consequences to it (check!), we have, up to a constant,
$$
V_\epsilon = \frac{\lambda}{4} \left (\left (\sigma'' + v (1+\epsilon/2) \right )^2 + \pi^2 -v^2\right )^2 -\epsilon \lambda v^3 \sigma'' \\
\approx \frac{\lambda}{4} (\sigma'' ^2 + \sigma '' v (2+\epsilon) +\epsilon v^2 + \pi^2)^2 -\epsilon \lambda v^3 \sigma '',
$$
which now displays a mass for the approximate goldston,
$$
m_\pi^2= \bbox[yellow,5px]{\epsilon\lambda v^2 /2 },
$$
while shifting the mass of the σ" to $m^2_{\sigma''}=\lambda v^2 (1+3\epsilon /2)$.
- The cognoscenti and lattice consumers might discern the flotsam of Dashen's formula here, $\bbox[yellow,5px]{\epsilon= m_q \Lambda^3/\lambda f_\pi^4}$ : the square of the pseudogoldston mass goes as the quark mass, not its square!
Note there is no linear term in σ" left over (no tadpoles).
Tilting the sombrero has given the Goldstone mode a minuscule σ" component, $\langle \delta \pi \rangle=v(1+\epsilon/2)$ and now $\langle \delta \sigma'' \rangle=-\langle \pi\rangle =O(\epsilon) $.
The symmetry current of O(2), $J_\mu= \sigma \partial_\mu \pi - \pi \partial_\mu \sigma$ is now evidently only Partially Conserved, on-shell,
$$
\partial \cdot J=\bbox[yellow,5px]{ \epsilon \lambda v^2 ~\pi }~.
$$
This is essentially the linchpin formula invented by Feynman and understood by Gell-Mann as PCAC, eqn (5) in the game-changing Gell-Mann, M., & Lévy, M. (1960), "The axial vector current in beta decay", Nuov Cim (1955-1965), 16(4), 705-726.
