“As you are probably already aware, census records are notoriously incorrect.” I don’t think they are “notoriously incorrect” but accept they can often/sometimes be unreliable.
So try another tack.
If the parents are Thomas Monks & Hannah Pryor, married Sheffield March quarter 1875, then consider these births (GRO):
MONKS, THOMAS PRYOR
GRO Reference: 1875 June Quarter in SHEFFIELD Volume 09C Page 475
MONKS, JOHN PRYOR
GRO Reference: 1878 December Quarter in SHEFFIELD Volume 09C Page 481
MONKS, WALTER PRYOR
GRO Reference: 1879 September Quarter in SHEFFIELD Volume 09C Page 485
MONKS, MARY ELLEN PRYOR
GRO Reference: 1883 March Quarter in SHEFFIELD Volume 09C Page 484
The 1879 Walter is probably this Walter of Lord Street in the burial records here:
MONKS, Walter (Son of Thomas Monks, age 4 months).
Died at Lord Street, Park; Buried on November 26, 1879 in General ground;
Grave Number 45, Section Q of Burngreave Cemetery, Sheffield.
If the Monks-Pryor parents are all the same couple, then an August 1879 birth of John would have been within months / weeks of Walter. This would not be possible unless they were twins. I see no evidence of that.
And another child death (same father, address & grave as Walter)
MONKS, Geo (Son of Thos Monks, age 3 months).
Died at Lord Street; Buried on January 2, 1881 in General ground;
Grave Number 45, Section Q of Burngreave Cemetery, Sheffield.