I would like to use the following phrase for a machine with large number cores. Is this correct? Please let me know.
large number core machine
"Large number core machine" sounds like clumsy or incorrect English. If you are talking about computers, the only grammatically-plausible interpretation that occurs to me is "Large number-core machine"; but that phrase is not semantically plausible because "number-core" is not a phrase used in the computer field.
Whether the previously-suggested many-core term is what you are looking for isn't clear. Some nuances noted in the wikipedia multi-core_processor description may rule it out: "A many-core processor is a multi-core processor in which the number of cores is large enough that traditional multi-processor techniques are no longer efficient... The many-core threshold is roughly in the range of several tens of cores... The terms many-core and massively multi-core are sometimes used to describe multi-core architectures with an especially high number of cores (tens or hundreds)."
Note that many-core refers to chips that contain many CPU cores, rather than to machines with many CPU's spread across many chips, which are multi-CPU machines.
If you have a specific chip in mind, with a specific number of cores, terms like dual-core, tri-core, quad-core, hexa-core, and octo-core are in common use.
If what you refer to is a GPU then look for terminology specific to such chips.
To answer the question directly, "Large number" is being used here as an adjective for core to express multiplicity. That is not the usual usage. As an adjective "Large number" typically refers to something related to numbers themselves, such as "large-number arithmetic".
According to Wikipedia: