Diffuse galaxy cluster emission at 168 MHz within the Murchison Widefield Array Epoch of Reionization 0-hour field

We detect and characterise extended, diffuse radio emission from galaxy clusters at 168 MHz within the Epoch of Reionization 0-hour field: a $45^circ times 45^circ$ region of the southern sky centred on R.~A.${}= 0^circ$, decl.${}=-27^circ$. We detect 29 sources of interest; a newly detected halo in Abell 0141; a newly detected relic in Abell 2751; 4 new halo candidates and a further 4 new relic candidates; and a new phoenix candidate in Abell 2556. Additionally, we find 9 clusters with unclassifiable, diffuse steep-spectrum emission as well as a candidate double relic system associated with RXC J2351.0-1934. We present measured source properties such as their integrated flux densities, spectral indices ($alpha$, where $S_ u propto u^alpha$), and sizes where possible. We find several of the diffuse sources to have ultra-steep spectra including the halo in Abell 0141, if confirmed, showing $alpha leq -2.1 pm 0.1$ with the present data making it one of the steepest-spectrum haloes known. Finally, we compare our sample of haloes with previously detected haloes and revisit established scaling relations of the radio halo power ($P_{1.4}$) with the cluster X-ray luminosity ($L_{mathrm{X}}$) and mass ($M_{500}$). We find that the newly detected haloes and candidate haloes are consistent with the $P_{1.4}$-$L_{mathrm{X}}$ and $P_{1.4}$-$M_{500}$ relations, and see an increase in scatter in the previously found relations with increasing sample size likely caused by inhomogeneous determination of $P_{1.4}$ across the full halo sample. We show that the MWA is capable of detecting haloes and relics within most of the galaxy clusters within the Planck catalogue of Sunyaev-Zeldovich sources depending on exact halo or relic properties.