I'm guessing that "compensation" means pecuniary damages, thus "matter of compensation" means private law (contract and tort). Is LCJ Hewart saying that the accused negligence's exceeds contractual and tortious negligence, and must be criminalized as criminal negligence?
The standard is that of gross rather than ordinary negligence. The difference between the two is a matter of degree and judgement in each case. Broadly speaking, negligence will be gross if the defendant’s conduct not merely fails to meet the standard set by the reasonable person test, but falls short of that standard by a considerable margin—i.e. if the defendant’s conduct is not merely unreasonable, but very unreasonable:196
“in order to establish criminal liability the facts must be such that, in the opinion of the jury, the negligence of the accused went beyond a mere matter of compensation between subjects and showed such disregard for the life and safety of others as to amount to a crime against the State and conduct deserving punishment.”
It may be negligent to drive around a particular bend at 50 mph; if so, it is grossly negligent to do so at 80 mph. Hart puts the test another way: “Negligence is gross if the precautions to be taken against harm are very simple, such as persons who are but poorly endowed with physical and mental capacities can easily take.”196
195 Bateman (1927) 19 Cr App R 8, 11–12. In manslaughter, this is a demanding standard: Sellu  EWCA Crim 1716,  4 WLR 64, –.
196 Punishment and Responsibility (1968) 149.
Simester and Sullivan's Criminal Law (2019 7 edn). p 168.
(a) The elements of gross negligence manslaughter
To cause or allow the death of another negligently will not, of itself, incur liability in manslaughter. The failure of care or concern must be more egregious than is required for civil liability. According to Lord Hewart CJ in Bateman, it must be the case that:330
“in the opinion of the jury, the negligence of the accused went beyond a mere matter of compensation between subjects and showed such disregard for the life and safety of others, as to amount to a crime against the state and conduct deserving of punishment.”
The dictum has been criticised for vacuity, yet it usefully stresses that juries should be cautious when attributing criminality to a defendant who has neither intended nor foreseen that she would cause or allow harm to another. The dictum was approved by the House of Lords in Andrews331 and reflects a consistent pattern of jury directions.
330 (1925) 19 Cr App R 8, 11.
331  AC 576, 582–3.
Op. cit. p 432.