The Galactic Cold Cores (GCC) project has made Herschel observations of interstellar clouds where Planck detected compact sources of cold dust emission. Our aim is to characterise the structure of the clumps and their parent clouds. We also examine the accuracy to which the structure of dense clumps can be determined from submillimetre data. We use standard statistical methods to characterise the GCC fields. Clumps are extracted using column density thresholding and we construct for each field a three-dimensional radiative transfer (RT) model. These are used to estimate the relative radiation field intensities, clump stability, and the uncertainty of column density estimates. We examine the radial column density profiles of the clumps. In the GCC fields, the structure noise follows the relations previously established at larger scales. The fractal dimension has no significant dependence on column density and the values D = 1.25 +- 0.07 are only slightly lower than in typical molecular clouds. The column density PDFs exhibit large variations, e.g. in the case of externally compressed clouds. At scales r>0.1 pc, the radial column density distributions of the clouds follow an average relation of N~r^{-1}. In spite of a great variety of clump morphology, clumps tend to follow a similar N~r^{-1} relation below r~0.1 pc. RT calculations indicate only factor of 2.5 variation in the local radiation field intensity. The fraction of gravitationally bound clumps increases significantly in regions with A_V > 5 mag but most bound objects appear to be pressure-confined. The GCC host clouds have statistical properties similar to general molecular clouds. The gravitational stability, peak column density, and clump orientation are connected to the cloud background while most other statistics (e.g. D and radial profiles) are insensitive to the environment.
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