If the circle in the sketch is the circular coil you are talking about, there will be an induced current if there is a time-varying magnetic flux through it.
The magnetic flux is defined as:
$$ \phi_B = \int_{\text{S bounded by loop}} \mathbf{B}\cdot\mathrm{d}\mathbf{S}. $$
Now, when you have a magnetic field in a material (and not just free space), you also need to take into account how the magnetism of the material itself may modify the total net field. The material, in this case, is whatever your coils is wrapped around. It's the material in the region with the 4 crosses inside the loop.
For this reason, you define $\mathbf{B}$ to be the net field in the region (external + material's response), and $\mathbf{H}$ to be the "magnetising field" i.e. the external field. The two are related by:
$$ \mathbf{B} = \mu \mathbf{H}, $$
where $\mu$ is the magnetic permeability (generally a rank-2 tensor, but let's assume a decent material so that it's a scalar here).
$\mu = \mu_0 \cdot \mu_{\mathrm{r}}$, where $\mu_0$ is the permeability of free space and $\mu_{\mathrm{r}}$ is the relative permeability of the material in question.
So all together now, the magnetic flux is:
$$ \phi_B = \mu_0 \int_S \mu_{\mathrm{r}} \mathbf{H}\cdot\mathrm{d}\mathbf{S}. $$
For an induced current, current, you need $\partial_t \phi_B \neq 0$. In order to do this, you can either vary the size of the cross section $\mathrm{d}\mathbf{S}_\parallel$ (by e.g. rotating the loop), vary the external field $\mathbf{H}$, or vary the relative permeability $\mu_{\mathrm{r}}$. Or all of them at the same time.
So if you (somehow) can control the external field's strength and the material's relative permeability independently, you can indeed keep $\mathbf{H}$ fixed and just vary $\mu_{\mathrm{r}}$ to get an induced current.
By the way, uniform magnetic field in this case means that it's only in one direction (into the paper). Provided $\mathbf{B}$ and $\mathbf{H}$ are parallel, i.e. when $\mu$ is a scalar and not a tensor, the field is uniform all the time.
